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 Simplify the Boolean function
Example 3.2 Simplify the Boolean Function
Three variable map 00 01 11 10 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 1
Three-variable map 00 01 11 10 1
Three-variable map 00 01 11 10 1
Example 3.3
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 1 Find adjacent squares
Problem 3.3 (a) Simplify the following Boolean function, using three-variable maps 00 01 11 10 1 Find adjacent squares
Problem 3.3 (a) Voila! Another solution (without a three-variable map) Factorize the expression Voila!
Four-Variable Map
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
Adjacency in a four-variable map
Adjacency in a four-variable map
Example 3.5 Simplify the Boolean Function 00 01 11 10 1 1 1 1 1 1 1 1
Example 3.6 Simplify the Boolean Function 00 01 11 10 1 1 1 1 1 1 1
Prime implicants When choosing adjacent squares in a map, make sure that: 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)
Example 3.5 (revisited) Simplify the Boolean Function 00 01 11 10 1 1
Example 3.6 (revisited) Simplify the Boolean Function 00 01 11 10 1 1
Prime implicants Simplify the function 00 01 11 10 1 1 1 1 1 1 1 1 1 1
Prime implicants Simplify the function 00 01 11 10 1 1 1 1 1 1 1 1 1 1
Prime implicants Simplify the function 00 01 11 10 1 1 1 1 1 1 1 1 1 1
Prime implicants Simplify the function 00 01 11 10 1 1 1 1 1 1 1 1 1 1
Five-Variable Map
Five-Variable Map How can adjacency be visualized in a five-variable map? 00 01 11 10
Five-Variable Map
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
NAND and NOR Implementation
Two-level implementation
NAND and NOR Implementation Implement the following Boolean function with NAND gates
Multilevel NAND circuits
Multilevel NAND circuits
NOR Implementation
NOR Implementation
NOR Implementation
NOR Implementation
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
Exclusive-OR implementations
Parity generation and checking
Parity generation and checking
Parity generation and checking