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E 1121 DIGITAL LOGIC (A)

FALL 2019

Lecture 1 Introduction & Number System

Course information

 Lecture Schedule Mon. 11.30AM – 1.10 PM

 Instructor

 Dr. Marwa Abdelrazik Elmenyawi  [email protected]  Office Hours: Monday 1.30 - 3.00 PM.

 Teaching Assistants

 Eng. Eman  Eng. Rehab  Eng. Mahmoud

 Course Link:

 piazza.com/benha_university_-_benha_faculty_of_engineering/fall 2019 /e 1121 /home

Course Syllabus

Course Objectives

 Explain the elements of digital system abstractions such as

 digital logic, Boolean algebra…… etc.

 Design simple digital systems based on these digital abstractions

 Use basic digital tools and devices such as FPGA and VHDL.

 Work in a design team that

 can propose, design, successfully implement, and report on a digital circuit design project.

Course Topics

 Number Systems

 Logic Gates

 Boolean Algebra

 Karnaugh Map

 Combinational circuits

 Boolean function, truth table, circuit  Decoder/Encoder  Multiplexer/Demultiplexer  Adder/Subracter/Multiplier

 Hardware Description Language

Chapter 1 Number Systems and Codes

• Computers and other digital systems process information as their primary function.
• It is necessary to have methods and systems for representing information in forms that can be manipulated and stored using electronic or other types of hardware.

Outline of Chapter 1

 Number Systems

 Number-base Conversions

 Arithmetic Operations

 Complements

 Signed Binary Numbers

 Binary Codes

Introduction to Numbering Systems  We are all familiar with the decimal number system (Base 10 ). Some other number systems that we will work with are:

 Binary

 Computers work only on two states on / off  Thus a number system with two elements {0,1}

 Octal

 Hexadecimal

1 Computer scientists are often looking for shortcuts to do things

Common Number Systems

System Base Basic digit

Used by

humans?

Used in

computers?

Decimal 10 0 , 1 , … 9 Yes No

Binary 2 0 , 1 No Yes

Octal 8 0 , 1 , … 7 No No

Hexadecimal 16 0 , 1 , … 9 ,

A, B, … F

No No

3

Characteristics of Numbering Systems

1) The digits are consecutive.

2) The number of basic digits is equal to the size of the base.

3) Zero is always the first digit.

4) The base number is never a digit.

5) When 1 is added to the largest basic digit, a sum of zero and a carry of one results.

6) Numeric values are determined by the implicit positional values of the digits.

4

Outline of Chapter 1

 Number Systems

 Number-base Conversions

 Arithmetic Operations

 Complements

 Signed Binary Numbers

 Binary Codes

Bridging the Digital Divide

Binary-to-Decimal Conversion Decimal-to-Binary Conversion 6

 A number expressed in base R can be converted to its decimal equivalent by using the

Positional representation rule [ multiplying each digit to R power of the digit’s place

and adding]

 N=

Base R to decimal conversion

Examples : 4 3 2 1 0 - 1 - 2 ( 11010. 1 1) 2 (? ) 10 N = 1  2 4

• 1  2 3
• 0  2 2
• 1  2 1
• 0  2 0
• 1  2

• 1
  • 1  2
    • 2 = (26.75) 10 2 1 0 - 1 (628.4) 9  (? ) 10 N = 6  9 2

• 2  9 1
• 8  9 0
• 4  9

• 1 = (512.1111) 10 2 1 0 - 1 ( 95 A.D) 16  (? ) 10 N = 9  16 2

• 5  16 1
• 10  16 0
• 13  16

• 1 = (2394.8125) 10 8

Converting Decimal to Base R

 To convert a decimal integer into base R, keep dividing by R until the quotient is 0.

Collect the remainders in forward order

 To convert a fraction, keep multiplying the fractional part by R until it becomes 0.

Collect the integer parts in reverse order

 Example: (162.375) 10 = (10100010.011) 2

162 81 0 40 1 20 0 10 0 5 0 2 1 1 0 0 1 0.375 x 2 = 0. 0.750 x 2 = 1. 0.500 x 2 = 1. 9