Logic Design and Digital Computer CircuitCMSC208

Percival A. Fernandez

Digital BasicsComponents of Digital BasicsBinary SystemsDigital Circuit TypesDesign TechniquesLogic OperationsNumber RepresentationsRepresentationsGatesKarnaugh Maps

Digital BasicsDigital vs Analog Signals

real world vs digital world Zero/low and One/high

Digital logicSwitching algebra (Boolean)Circuits to realize Boolean functions (gates)

Digital BasicsAnalog versus Digital ElectronicsMost (but not all) observables are analog

◦think of analog versus digital watchesBut the most convenient way to

represent and transmit information electronically is digital◦think of audio recordings vs original Edison wax

recordingsAnalog/Digital and Digital/Analog

conversion is essential (and nothing new)◦think of a piano keyboard vs violin

Logical Operations and Truth Tables

At first glance, it may not seem that the study of logic should be part of mathematics. For most of us, the word logic is associated with reasoning in a very nebulous way.

The purpose of logic is to enable the logician to construct valid arguments which satisfy the basic principle

Logical Operations

A logical variable can only have the value True or False. That is, True and False are the "numerical constants" of logic. And instead of the usual arithmetic operators (addition, subtraction, etc.), the logical operators are "AND", "OR", "NOT", "XOR" ("eXclusive OR"), "IMPLIES" and "EQUIVALENCE".

Truth Tables

In logic, we only have two "numbers": True and False. Therefore, any logical statement which contains a finite number of logical variables (which of course covers any problem we have to deal with) can be analyzed using a table which lists all possible values of the variables: a "truth table".

Truth TablesSince each variable can take only two

values, a statement with "n" variables requires a table with 2n rows. Using the letters "p", "q", "r", etc., to represent logical variables, we can construct truth tables for statements involving any number of variables (although we will usually limit ourselves to at most three variables per statement to simplify the matter): p

TF

Truth Tables• for statements with one variable,

p q rT T TT T FT F TT F FF T TF T FF F TF F F

Logic Operations – Logic Basics

Logic operations include any operations that manipulate Boolean values. Boolean values are either true or false.

They are named after English mathematician George Boole, who invented Boolean algebra, and is widely considered the founder of computer science theory. They can also be represented as 1 and 0. Normally, 1 represents true, and 0 represents false, but it could be the other way around.

The basic Boolean operators are and (^), or (v), and not ('). Boolean functions can be built from these three basic operators.

Truth Tables Given two

Boolean variables A and B, the Boolean expression A ^ B is true only if both A and B are true. The truth table below illustrates all possible combinations with the AND operator.

A B A ^ B

0 0 0

0 1 0

1 0 0

1 1 1

Table 1. The AND Operator

Truth Tables Not A is true

when A is false, and vice-versa.

Table 1. The NOT Operator

A A'0 11 0

The End

References: http://www.electronicsteacher.com/computer-

architectures/digital-circuits/combinatorial.php http://www.buzzle.com/articles/different-types-of-

computers.html Structured Computer Organization, Andrew S.

Tanembaum