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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
3.7
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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:
3.9
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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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