Spring 2015 Week 1 odule 4 Digital Circuits and Sstems interms, aterms SoP and PoS forms Shankar Balachandran* Associate Professor, CSE Department Indian Institute of Technolog adras *Currentl a Visiting Professor at IIT Bomba
Some Definitions A literal is a complemented or uncomplemented boolean variable. Eamples: a and ā are distinct literals. ā+cd is not. A product term is a single literal or a logical product (AND) of two or more literals. Eamples: a, ā, ac, ācd, aaāb are product terms; ā+cd is not a product term. A sum term is a single literal or a logical sum (OR) of two or more literals. Eamples: a, ā, a+c, ā+c+d are sum terms; ā+cd is not a sum term. Digital Logic Fundamentals 2
Some Definitions A normal term is a product or sum term in which no variable appears more than once. Eamples: a, ā, a+c, ācd are normal terms; ā+a, āa are not normal terms. A minterm of n variables is a normal product term with n literals. There are 2 n such product terms. Eamples of 3-variable minterms: ābc, abc Eample: āb is not a 3-variable minterm. A materm of n variables is a normal sum term with n literals. There are 2 n such sum terms. Eamples of 3-variable materms: ā+b+c, a+b+c Digital Logic Fundamentals 3
Some Definitions A sum of products (SOP) epressions is a set of product (AND) terms connected with logical sum (OR) operators. Eamples: a, ā, ab+c, āc+bde, a+b are SOP epressions. A product of sum (POS) epressions is a set of sum (OR) terms connected with logical product (OR) operators. Eamples: a, ā, a+b+c, (ā+c)(b+d) are POS epressions. Digital Logic Fundamentals 4
Some Definitions The canonical sum of products (CSOP) form of an epression refers to rewriting the epression as a sum of minterms. Eamples for 3-variables: ābc + abc is a CSOP epression; āb + c is not. The canonical product of sums (CPOS) form of an epression refers to rewriting the epression as a product of materms. Eamples for 3-variables: (ā+b+c)(a+b+c) is a CPOS epression; (ā+b)c is not. There is a close correspondence between the truth table and minterms and materms. Digital Logic Fundamentals 5
Deorgan s Theorem (revisited) X X... X X X... X 1 2 n 1 2 n X X... X X X... X 1 2 n 1 2 n Complement of Sum of Products is equivalent to Product of Complements. Complement of Product of Sums is equivalent to Sum of Complements. Digital Logic Fundamentals 6
interms A minterm can be defined as a product term that is 1 in eactl one row of the truth table. n variable minterms are often represented b n- bit binar integers. How to associate minterms with integers? State an ordering on the variables Form a binar number Set bit i of the binar number to 1 if the i th the minterm in an uncomplemented form variable appears in Set bit i to 0 if the variable appears in the complemented form. Digital Logic Fundamentals 7
interm Eamples Assume a 3-variable epression, F,, min term 000 min term 011 min term111 m m m 000 011 111 m 0 m m 3 7 F,, m 0 m 3 m0 0, 3, 7,m m 3,m 7 7 Digital Logic Fundamentals 8
aterms A materm can be defined as a sum term that is 0 in eactl one row of the truth table. n variable materms are also represented b n- bit binar integers. How to associate materms with integers? State an ordering on the variables Form a binar number Set bit i of the binar number to 0 if the i th the materm in an uncomplemented form variable appears in Set bit i to 1 if the variable appears in the materm in the complemented form. Digital Logic Fundamentals 9
Digital Logic Fundamentals 10 aterm Eamples Assume a 3-variable epression,,, F 4 100 1 001 0 000 100 001 000 ma term ma term ma term 4 0 1 4 1 0 4 1 0,,,,,, F
Summar of interms and aterms Digital Logic Fundamentals 11
A Sample Three Variable Function f,, m, m, m, m 1 2 3 1 4 5 6 1,4,5,6 f,,,,, 1 2 3 0 2 3 7 0,2,3,7 Digital Logic Fundamentals 12
2 f 3 1 (a) A minimal sum-of-products realiation 1 3 f 2 (b) A minimal product-of-sums realiation
End of Week 1: odule 4 Thank You Digital Logic Fundamentals 14