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Gate-Level Minimization. Chapter 3. The Map Method. Two-variable map. Two-variable map. Three-variable map. Adjacent. Adjacent. Adjacent when minterms differ by one variable. Graphical view of adjacency. Taking the difference between adjacent squares gives:. Example 3.1.
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Gate-Level Minimization Chapter 3
The Map Method Two-variable map
Three-variable map Adjacent Adjacent Adjacent when minterms differ by one variable
Graphical view ofadjacency Taking the difference between adjacent squares gives:
Example 3.1 Simplify the Boolean function
Example 3.2 Simplify the Boolean Function
Three variable map 00 01 11 10 0 1
Three-variable map • The number of adjacent squares that may be combined must always represent a number that is a power of two: • One square represents one minterm which in this case gives a term with three literals • Two adjacent squares represent a term with two literals • Four adjacent squares represent a term with one literal • Eight adjacent squares encompass the entire map and produce a function equal to 1
Three-variable map 00 01 11 10 0 1
Three-variable map 00 01 11 10 0 1
Three-variable map 00 01 11 10 0 1
Example 3.4 Let the Boolean Function (a) Express this function as a sum of minterms
Example 3.4 (b) Find the minimal sum-of-products expression 00 01 11 10 0 1 Find adjacent squares
Problem 3.3 (a) Simplify the following Boolean function, using three-variable maps 00 01 11 10 0 1 Find adjacent squares
Problem 3.3 (a) Another solution (without a three-variable map) Factorize the expression Voila!
Four-variable map • The number of adjacent squares that may be combined must always represent a number that is a power of two: • One square represents one minterm which in this case gives a term with four literals • Two adjacent squares represent a term with three literals • Four adjacent squares represent a term with two literals • Eight adjacent squares represent a term with one literal • Sixteen adjacent squares encompass the entire map and produce a function equal to 1
Example 3.5 Simplify the Boolean Function 00 01 11 10 00 1 1 1 01 1 1 1 11 1 1 1 1 1 10
Example 3.6 Simplify the Boolean Function 00 01 11 10 00 1 1 1 01 1 11 10 1 1 1
Prime implicants • All the minterms are covered when combining the squares • The number of terms in the expression is minimized • There are no redundant terms (minterms already covered by other terms) When choosing adjacent squares in a map, make sure that:
Example 3.5 (revisited) Simplify the Boolean Function 00 01 11 10 00 1 1 1 01 1 1 1 11 1 1 1 1 1 10
Example 3.6 (revisited) Simplify the Boolean Function 00 01 11 10 00 1 1 1 01 1 11 10 1 1 1
Prime implicants Simplify the function 00 01 11 10 00 1 1 1 01 1 1 11 1 1 1 1 1 10 1
Prime implicants Simplify the function 00 01 11 10 00 1 1 1 01 1 1 11 1 1 1 1 1 10 1
Prime implicants Simplify the function 00 01 11 10 00 1 1 1 01 1 1 11 1 1 1 1 1 10 1
Prime implicants Simplify the function 00 01 11 10 00 1 1 1 01 1 1 11 1 1 1 1 1 10 1
Five-Variable Map How can adjacency be visualized in a five-variable map? 00 01 11 10 00 01 11 10
Five-Variable Map Simplify the Boolean function
Product-Of-Sums Simplification Take the squares with zeros and obtain the simplified complemented function Complement the above expression and use DeMorgan’s
Product-Of-Sums Simplification Sum-of-products Product-of-sums
Don’t-Care Conditions • Used for incompletely specified functions, e.g. BCD code in which six combinations are not used (1010, 1011, 1100, 1101, 1110, and 1111) • Those unspecified minterms are neither 1’s nor 0’s • Unspecified terms are referred to as “don’t care” and are marked as X • In choosing adjacent squares, don’t care squares can be chosen either as 1’s or 0’s to give the simplest expression
Don’t-Care Conditions Simplify the Boolean function which has the don’t care conditions
NAND and NOR Implementation Implement the following Boolean function with NAND gates
Exclusive OR Function Exclusive-OR or XOR performs the following logical operation Exclusive-NOR or equivalence performs the following logical operation Identities of the XOR operation XOR is commutative and associative