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18/19/2011 EEE C391/ECE C391/CS C391 18/19/2011

Digital Electronics and ComputerOrganization

Instructor

Mr. Sai Krishna PSProf. Moorthy Muthukrishnan

Lectures: M-W-F - 3hrs

Tutorial: Saturday-1hr

Lecture 6,7

Simplification of Boolean Functions

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Gate Level minimization

Why is Minimization Necessary: Difficulty in manual methods to minimize

when a logic circuit has more than few inputs.

To obtain an optimal gate level implementation.

The Map method:

Simple and straightforward method for minimizing a Boolean

function, pictorial form of a truth table.

Known as Karnaugh map or K-map

Diagram made up of squares,each square representing one min-term of the function

A Boolean function can be represented in each of these squares

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Visual diagram of a function in standard form.

Alternate algebraic functions by visualizing the various functions

Select the simplest of all functions

Simplified expression in one of the two standard forms

Sum of products

Product of sums

Minimum number of terms,the smallest possible number of literals in each term

Produces a circuit diagram withminimum number of gatesminimum number of inputs to each gate.

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The K-Map method

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The K-Map method

Two-variable K-map:

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The K-Map method

Example of a two-variable K-map:

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The K-Map method

Three variable K-map: yz

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The K-Map method

Example of a three variable K-map:

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The K-Map method

Example of a three variable K-map:

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The K-Map method

Example of a three variable K-map:

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The K-Map method

Example of a three variable K-map:

Problem Solution

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The K-Map method

Four variable K-map:

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The K-Map method

Example of a Four variable K-map:

1

3.6

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The K-Map method

Example of a Four variable K-map:

Problem

Solution

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Note :When choosing adjacent Squares in a map

one must ensure that:

1. All the min-terms of the function are covered,

when we combine the squares.

2. The number of terms in the expression is minimized.

3. There are no redundant terms

i.e. min-terms already covered by other terms.

Also there can be two or more expressions

that satisfy the simplification criteria.

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Prime Implicants and Essential PrimeImplicants

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Prime Implicant: Product term obtained by combining

the maximum possible number of adjacent squares in the map.

If a minterm in a square is covered by

only one prime implicant that prime implicantis said to be ESSENTIAL

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Prime Implicants andEssential Prime Implicants

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Prime Implicants andEssential Prime Implicants

BD, BD are

essential primeimplicants

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The K-Map method

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Example of a Five variable K-map:

Adjacent

Adjacent

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The K-Map method

Example of a Five variable K-map:

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The K-Map method

Example of a Four variable POS Simplification:

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Dont care Conditions:

Function is not specified under certain combination of variables

incompletely specified functions.

The unspecified min-terms are called Dont care conditions.

Logical value cannot be marked as 1 or 0 in the mapX is used.

Dont care min-terms may be assumed as either 0 or 1

for simplifying the expression.

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The K-Map method

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The K-Map method

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