Digital logic COMP214
 Lecture 1
Dr. Sarah M.Eljack
Chapter 1
1
‫مقرر المنطق الرقمي‬
1. Concepts and basic components of computer
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hardware and software.
2. Systems numbers
3. binary Systems
4. Boolean algebra
5. Logical expressions
6. Logic gates
7. Digital circuit design Structural
8. Sequential digital circuits
Chapter 1
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References
:‫المراجع‬
 1. Digital Design, Mano, M. Morris, 3rd
edition, ISBN 0-13-062121-8, PrenticeHall, 2002.
 2. Fundamentals of logic Design, 5th edition,
Brooks/Cole Thomson Learning, 2004.
 3. Introduction to Digital Systems, M.
Ercegovac, T. Lang, and J. Moreno, Wiley
and Sons. 1998.
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Chapter 1
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Outline
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Introduction
Numbering Systems
Binary & Hexadecimal Numbers
Base Conversions
Integer Storage Sizes
Binary and Hexadecimal Addition
Signed Integers and 2's Complement Notation
Binary and Hexadecimal subtraction
Carry and Overflow
Character Storage
Overview
 Digital Systems and Computer Systems
 Information Representation
 Number Systems [binary, octal and hexadecimal]
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Digital System
 Takes a set of discrete information inputs and discrete
internal information (system state) and generates a set
of discrete information outputs.
Discrete
Inputs
Discrete
Information
Processing
System
Discrete
Outputs
System State
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Digital System Example:
A Digital Counter (e. g., odometer):
Count Up
Reset
0 0 1 3 5 6 4
Inputs: Count Up, Reset
Outputs: Visual Display
"Value" of stored digits
State:
Synchronous or Asynchronous?
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A Digital Computer Example
Memory
CPU
Inputs:
Keyboard,
mouse, modem,
microphone
Control
unit
Datapath
Input/Output
Outputs: CRT,
LCD, modem,
speakers
Synchronous or
Asynchronous?
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Signal
 An information variable represented by physical
quantity.
 For digital systems, the variable takes on discrete
values.
 Two level, or binary values are the most prevalent
values in digital systems.
 Binary values are represented abstractly by:
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•
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digits 0 and 1
words (symbols) False (F) and True (T)
words (symbols) Low (L) and High (H)
and words On and Off.
 Binary values are represented by values or ranges of
values of physical quantities
Chapter 1
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Signal Examples Over Time
Time
Analog
Digital
Asynchronous
Synchronous
Continuous
in value &
time
Discrete in
value &
continuous
in time
Discrete in
value & time
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Signal Example – Physical Quantity: Voltage
OUTPUT
INPUT
5.0
HIGH
4.0
3.0
2.0
LOW
1.0
HIGH
Threshold
Region
LOW
0.0
Volts
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Binary Values: Other Physical Quantities
 What are other physical quantities
represent 0 and 1?
• CPU Voltage
• Disk Magnetic Field Direction
• CD Surface Pits/Light
• Dynamic RAM Electrical Charge
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Introduction
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Computers only deal with binary data (0s and 1s), hence all data
manipulated by computers must be represented in binary format.
Machine instructions manipulate many different forms of data:
• Numbers:
 Integers: 33, +128, -2827
 Real numbers: 1.33, +9.55609, -6.76E12, +4.33E-03
• Alphanumeric characters (letters, numbers, signs, control
characters): examples: A, a, c, 1 ,3, ", +, Ctrl, Shift, etc.
• Images (still or moving): Usually represented by numbers
representing the Red, Green and Blue (RGB) colors of each pixel in
an image,
• Sounds: Numbers representing sound amplitudes sampled at a certain
rate (usually 20kHz).
So in general we have two major data types that need to be represented in
computers; numbers and characters.
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Number Systems – Representation
 Positive radix, positional number systems
 A number with radix r is represented by a
string of digits:
An - 1An - 2 … A1A0 . A- 1 A- 2 … A- m + 1 A- m
in which 0 Ai < r and . is the radix point.
 The string of digits represents the power series:
(
i=n-1
(Number)r =
i=0
Ai
r )+( 
j=-1
i
Aj
r)
j
j=-m
(Integer Portion) + (Fraction Portion)
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.
 Numbering systems are characterized by
their base number.
 In general a numbering system with a base r
will have r different digits (including the 0)
in its number set. These digits will range
from 0 to r-1
 The most widely used numbering systems
are listed in the table below:
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Chapter 1
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Number Systems – Examples
Radix (Base)
Digits
0
1
2
3
Powers of 4
Radix
5
-1
-2
-3
-4
-5
General
Decimal
Binary
r
10
2
0 => r - 1
0 => 9
0 => 1
r0
r1
r2
r3
r4
r5
r -1
r -2
r -3
r -4
r -5
1
10
100
1000
10,000
100,000
0.1
0.01
0.001
0.0001
0.00001
1
2
4
8
16
32
0.5
0.25
0.125
0.0625
0.03125
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Special Powers of 2
 210 (1024) is Kilo, denoted "K"
 220 (1,048,576) is Mega, denoted "M"
 230 (1,073, 741,824)is Giga, denoted "G"
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Binary Numbers
1 1 1 1 1 1 1 1
 Each digit (bit) is either 1 or 0 2 2 2 2 2 2 2
 Each bit represents a power of 2
 Every binary number is a sum of powers of
2
7
6
5
4
3
2
1
20