Fall 2012: FCM 708 Foundation I Lecture 2 Prof. Shamik Sengupta

Quick Recap…  Intro to Computer Architecture: –Number system –Decimal, Binary, Hexadecimal –Unsigned and signed representations –Hardware architecture –A simplified model of the microprocessor structure –Central Processing Unit (CPU) –Arithmetic & Logic Unit (ALU) –Control Unit (CU) –Register Array –System Bus –Memory –Overview of Instruction Execution Cycle FCM 708: Sengupta

A quick look at a microprocessor architecture  Let us have some hand-on experience of what we have learnt so far  We will use a simple microprocessor simulator –Motorola 68HC11 FCM 708: Sengupta

Boolean algebra and Logic gates

Objectives  Understand the relationship between Boolean logic and digital computer circuits  Learn how to design simple logic circuits.  Understand how digital circuits work together to form complex computer systems. FCM 708: Sengupta

Introduction  In the latter part of the nineteenth century, George Boole showed that logical thought could be represented through mathematical equations  Computers, as we know them today, are implementations of Boole’s Laws of Thought –John Atanasoff and Claude Shannon were among the first to see this connection FCM 708: Sengupta

What is Boolean algebra  Boolean algebra is an algebra for the manipulation of objects that can take on only two values, typically true and false  Why Boolean algebra is so useful in computers? –Because computers are built as collections of gates that are either “on” or “off,” Boolean algebra is a very natural way to represent digital information or compute information  Boolean functions are implemented in digital computer circuits called gates (logic gates) –A gate is an electronic device that produces a result based on two or more input values –All the microprocessor components are combinations of such logic gates FCM 708: Sengupta

Boolean Operators  Most common Boolean operators are AND, OR and NOT  A Boolean operator can be completely described using a truth table  The truth table for the Boolean operators AND, OR and NOT are shown here FCM 708: Sengupta

 The three simplest gates are the AND, OR, and NOT gates.  They correspond directly to their respective Boolean operations, as you can see by their truth tables  And these representations map exactly into the electric circuits of a digital system Logic Gates FCM 708: Sengupta

Detailed implementation picture of a Logic Gate Voltage inverted from input Voltage from input This is the logic for an AND gate 74LS08 Quad 2-input AND FCM 708: Sengupta

 The output of the XOR operation is true only when the values of the inputs differ. Logic Gates Note the special symbol  for the XOR operation. Symbols for NAND and NOR, and truth tables are shown at the right. FCM 708: Sengupta  Three other logic gates:

Logic Gates  NAND is known as universal gate because they are inexpensive to manufacture and any Boolean function can be constructed using only NAND gates. FCM 708: Sengupta

Boolean Functions  Boolean functions are composed of Boolean variables and multiple logic operators  NOT has the precedence over AND  AND has the precedence over OR FCM 708: Sengupta

Boolean Functions  Digital computers contain circuits that implement Boolean functions.  The simpler that we can make a Boolean function, the smaller the circuit that will result. –Simpler circuits are cheaper to build, consume less power, and run faster than complex circuits.  With this in mind, we always want to reduce our Boolean functions to their simplest form. –Boolean identities FCM 708: Sengupta

 Most Boolean identities have an AND (product) form as well as an OR (sum) form.  We show our identities using both forms. Our first group is rather intuitive: Boolean identities FCM 708: Sengupta

 Our second group of Boolean identities should be familiar to you from your study of algebra: Boolean identities FCM 708: Sengupta

 Our last group of Boolean identities are perhaps the most useful. Boolean identities FCM 708: Sengupta

Simplification of Boolean Functions  Let’s try some of these identities to simplify Boolean Functions:  F = AB + BBC + BCC  F = A + B(A+C) + AC FCM 708: Sengupta

 Simplify the function: Simplification of Boolean Functions FCM 708: Sengupta

Hand-on Practice  Multimedia Logic Simulator  Can be downloaded from  We will implement some of the simplest logic circuits FCM 708: Sengupta

Digital Circuits and Boolean Algebra  Using Boolean algebra to design various important digital circuits implementation –Designing a Burglar alarm –Designing an adder FCM 708: Sengupta

Reading Assignment 1.Boolean Algebra (In Blackboard) FCM 708: Sengupta