ECE 301 – Digital Electronics-Lecture_3.pptVIP

* * * * * * * * * * * * * * * ECE 301 - Digital Electronics * Simplification A + AB = A + B F = (XY + Z).(YW + ZV) + (XY + Z) F = YW + ZV + (XY + Z) A.(A + B) = A . B G = (X + Y).( (X + Y) + (WZ) ) G = (X + Y) . WZ ECE 301 - Digital Electronics * Logic Adjacency (Combining) A.B + A.B = A F = (X + Y).(WXZ) + (X + Y).(WXZ) F = (X + Y) (A + B).(A + B) = A G = (XY + XZ).(XY + (XZ) ) G = XY ECE 301 - Digital Electronics * Boolean Algebra Example: Using Boolean Algebra, simplify the following Boolean expression. F(A,B,C) = A.B.C + A.B.C + A.B.C ECE 301 - Digital Electronics * Boolean Algebra Example: Using Boolean Algebra, simplify the following Boolean expression. F(A,B,C) = (A+B+C).(A+B+C).(A+B+C) ECE 301 - Digital Electronics * DeMorgans Laws Can be stated as follows: The complement of the product (AND) is the sum (OR) of the complements. (X.Y) = X + Y The complement of the sum (OR) is the product (AND) of the complements. (X + Y) = X . Y Easily generalized to n variables. Can be proven using a Truth table ECE 301 - Digital Electronics * Proving DeMorgans Law (X . Y) = X + Y ECE 301 - Digital Electronics * x 1 x 2 x 1 x 2 x 1 x 2 x 1 x 2 x 1 x 2 x 1 x 2 x 1 x 2 x 1 x 2 + = (a) x 1 x 2 + x 1 x 2 = (b) DeMorgans Theorems ECE 301 - Digital Electronics * Importance of Boolean Algebra Boolean Algebra is used to simplify Boolean expressions. Through application of the Laws and Theorems discussed Simpler expressions lead to simpler circuit realization, which, generally, reduces cost, area requirements, and power consumption. The objective of the digital circuit designer is to design and realize optimal digital circuits. ECE 301 - Digital Electronics * Algebraic Simplification Justification for simplifying Boolean expressions: Reduces the cost associated with realizing the expression using logic gates. Reduces the area (i.e. silicon) required to fabricate the switching function. Reduces the power consumption of the c