Presentation on theme: "Gate-level Minimization"— Presentation transcript:
1 Gate-level Minimization Although truth tables representation of a function is unique, it can be expressed algebraically in different formsThe procedure of simplifying Boolean expressions (in 2-4) isdifficult since it lacks specific rules to predict the successive steps in the simplification process.Alternative: Karnaugh Map (K-map) Method.Straight forward procedure for minimizing Boolean FunctionFact: Any function can be expressed as sum of mintermsK-map method can be seen as a pictorial form of the truth table.yyxm0m1m2m3xTwo-variable map
3 Two-variable K-MAP y y y y x x x x The three squares can be determined from the intersectionof variable x in the second row and variable y in the secondcolumn.
4 Three-Variable K-MapAny two adjacent squares differ by only one variable.M5 is row 1 column 01. 101= xy’z=m5Since adjacent squares differ by one variable (1 primed, 1 unprimed)From the postulates of Boolean algebra, the sum of two minterms in adjacent squares can be simplified to a simple ANDFor example m5+m7=xy’z+xyz=xz(y’+y)=xz
11 Three-variable K-Map: Observations One square represents one minterm a term of 3 literalsTwo adjacent squares a term of 2 literalsFour adjacent squares a term of 1 literalEight adjacent squares the function equals to 1
16 Prime ImplicantsNeed to ensure that all Minterms of function are coveredBut avoid any redundant terms whose minterms are already coveredPrime Implicant is product Term obtained by combining maximum possible number of adjacent squaresIf a minterm in a square is covered by only prime implicant then ESSENTIAL PRIME IMPLICANTEssential prime implicant BD and B’D’Non Essential prime implicant CD, B’C, AD and AB’
17 Four-variable K-Map: Observations One square represents one minterm a term of 4 literalsTwo adjacent squares a term of 3 literalsFour adjacent squares a term of 2 literalEight adjacent squares a term of 1 literalsixteen adjacent squares the function equals to 1
19 SUM of PRODUCT and PRODUCT OF SUM Simplify the following Boolean function in:(a) sum of products (b) product of sumsCombining the one’s:(a)Combining the zero’s:Taking the the complement:(b)
20 SOP and POS gate implementation SUM OF PRODUCT (SOP)PRODUCT OF SUM (POS)
21 Implementation of Boolean Functions Draw the logic diagram for the following function: F = (a.b)+(b.c)abFc
22 Implement a circuit using OR and Inverter Gates only 2 LevelMore than two levelSOPPOSImplement a circuit using OR and Inverter Gates onlyImplement a circuit using AND and Inverter Gates onlyImplement a circuit using NAND Gates onlyImplement a circuit using NOR Gates only
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