INTRODUCTION

TO

DIGITAL ELECTRONICS

(Computer Hardware) Digital Signal Lecture 3 Lecture 4 Flip Flops By Engr. D. Garcia Digital electronics, or digital (electronic) circuits, represent signals by discrete bands of analog levels, rather than by a continuous range. ANALOG SIGNAL

- continuous

- Infinite set of value

- can easily affected by noise

- high resolution DIGITAL SIGNAL

- discrete values

- switches between two voltage levels to represent the two states of Bolean Value "1" and "0".

- noise Immunity

- signal regeneration BINARY SYSTEM (BASE 2) DECIMAL SYSTEM (BASE 10) 0 1 2 3 4 5 6 7 8 9 1 0 ON OFF

HIGH LOW

CLOSE OPEN

PULL PUSH

TRUE FALSE 0 0

1 1

2 10

3 11

4 100

5 101

6 110

7 111

8 1000

9 1001 Counting

in Base 10 Counting

in Base 2 Binary Counting 10^7 10^6 10^5 10^4 10^3 10^2 10^1 10^0 10,000,000 1,000,000 100,000 10,000 1,000 100 10 1 1 2 3 4 5 6 7 8 TWELVE MILLION THREE HUNDRED FORTY FIVE THOUSAND SIX HUNDRED SEVENTY EIGHT THIS IS EQUAL TO:

1X10,000,00 +

2X1,000,000 +

3X100,000 +

4X10,000 +

5X1000 +

6X100 +

7X10 +

8X1 BASE 10 BASE 2 2^7 2^6 2^5 2^4 2^3 2^2 2^1 2^0 128 64 32 16 8 4 2 1 1 0 0 1 0 1 1 1 = 128+0+0+16+0+4+2+1

= 151 IN YOUR POINT OF VIEW

WHY COMPUTERS USES BINARY? Some reasons are:

1.) Binary switches and devices that show two states either ON or OFF is much easier to build and cheap.

2.) Creating a computer higher than Binary might be very expensive since it must be very sensitive to store different value/states.

3.) You will be needing a very stable current to supply your computer.

4.) Transmitting data in a long distance will diminish the current. Signal regeneration is much more effective in binary. Boolean algebra (logic) Developed by

George Boole 1854 -the algebra of truth values 0 and 1 -based on two-valued or binary systems Binary systems

were known in the ancient Chinese civilization and by the classical Greek philosophers who created a well structured binary system, called propositional logic. Propositions may be TRUE or FALSE, and are stated as functions of other propositions which are connected by the three basic logical connectives: AND, OR, and NOT. For example the statement: “I will take a raincoat with me if it is raining or the sky is cloudy." OR Rain Cloudy Sky Raincoat Cloudy Raining Umbrella

False False False

False True True

True False True

True True True “If I do not take the car then I will take the raincoat if it is raining or the atmosphere is cloudy” Rain Cloudy Sky Car Raincoat

False False False

False False True

False True False

False True True

True False False

True False True

True True False

True True True OR Rain Cloudy

Sky NOT CAR AND Take Raincoat (Take Raincoat (U)) = ( NOT (Take Car (C) ) ) AND ( (Cloudy Sky (S) ) OR (Raining (R) ) ) 1.) Logic Operation Precedence

( ) Parenthesis - Highest

' NOT

. AND

+ OR - Lowest

2.) NOT Operator

(A')' = A (Double Negation or Involution

law,)

A . A' = 0 (Complement)

A + A' = 1 (Complement)

3.) Associative

(A B) C = A (B C)

(A + B) + C = A + (B + C)

4.) Commutative

A B = B A

A + B = B + A

5.) Distributive

A (B + C) = A B + A C

A + (B C) = (A + B) (A + C)

6.) Indempotent

A A = A

A + A = A

7.) Annulment

A 0 = 0

A + 1 = 1

8.) Identity

A , 1 = A

A + 0 = A

9.) de Morgan’s theorem

(A + B)' = A' B'

(A B)' = A' + B'

Ex. Simplifying

A + A (B) = A (proof: A + A B = A (1 + B) = A 1 = A)

A (A + B) = A (proof: A A + A B = A + A B = A ) Rules in Boolean Expressions U = C' . (S+R) Logic Gates Circuit Graphic Symbol Relay Transistor Boolean Function Truth Table Buffer X = A Inverter

(NOT) X = A' AND X=AB OR X = A + B NAND X = (AB)' NOR X = (A+B)' Exclusive OR

XOR Exclusive NOR

XNOR These gates are the fundamental building blocks of digital circuits. In fact it would be possible to build any digital circuit using just NAND gates or just NOR gates. NOR and NAND gate

Building Blocks of Digital Circuits NOT Gate Indempotent law:

A.A = A therefore (A.A)' = A' To create an AND gate we apply the Involution

law,

(A')' = A

making use of our newly designed inverter: AND gate OR gate To make an OR gate we need to apply de

Morgan's theorem:

A+B = (A' . B')' SEAT WORK

Can you build every gate you need from the

NOR gate? Source: http://www.doc.ic.ac.uk

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# Introduction to Digital Electronics

Introduction to Digital Electronics in Preparation for Computer Fundamentals (Hardware)

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