Digital Circuits and Systems Algebraic Minimization Examples - PowerPoint PPT Presentation

Spring 2015 Week 1 Module 5 Digital Circuits and Systems Algebraic Minimization Examples Shankar Balachandran* Associate Professor, CSE Department Indian Institute of Technology Madras *Currently a Visiting Professor at IIT Bombay

Summary of Minterms and Maxterms Algebraic Minimization Examples 2

Conversion between SOP and POS  Conversion between  and P representations is easy. Assume an n- variable function so the minterm and maxterm lists that represent the function are subsets of {0,1,……,2 n – 1}. It can be shown that the minterm indices and maxterm indices are complementary.  M m That is, i i  Example : Assume a 3 variable expression F(x,y,z).       1 , 4 , 7  0 , 2 , 3 , 5 , 6        m m m M M M M M 1 4 7 0 2 3 5 6 Algebraic Minimization Examples 3

Algebraic Simplification  To reduce circuit complexity and to maximize circuit performance, it is often necessary to write algebraic expressions in SOP or POS forms.  The rules discussed earlier are used to do the simplification. Algebraic Minimization Examples 4

Example:  Simplify to SOP form:     y      F x , y , z x y z z      x y z y z y    x y y x z y z y     x y x z y z y 1      x y z y x 1    x y z y 1   x y z y     F x , y , z x y z y Algebraic Minimization Examples 5

Example:  Write the following to canonical SOP (sum of minterms) form.     f x , y , z x y z y         x y z z z y x x     x y z x y z x y z x y z    x y z x y z x y z    x y z x y z x y z      f x , y , z  2 , 3 , 6 Algebraic Minimization Examples 6

Example:  Simplify to POS form:      f x , y , z x y z x y x y z      x y z z x y   x y x y     x x y   1 y  y Algebraic Minimization Examples 7

Example:  Simplify to SOP and POS forms.           a b a b c d b c c c d a b c b c d     ab abcd bc 0      ab cd bc 1   ab bc ...... SOP form     b a c ...... POS form Algebraic Minimization Examples 8

Example:  Simplify to POS and expand to canonical POS (product of maxterms).     f x , y , z x y x z        x y x x y z            x x y x x z y z          x y x z y z ........ POS form            x y z z x y y z x x y z                         x y z x y z x y z x y z x y z x y z                x y z x y z x y z x y z       0 2 4 5 f x,y,z , , , ...... Canonical POS form Algebraic Minimization Examples 9

Example:  Simplify to SOP form:           w x y z wxz    w x y z wxz           w x y z w x z        w x y z w x z          w y x y z w x z        w x w y x y y y z w x z        w x z w y z x y z y z w x z          w x z z y z w x 1   w x y z Algebraic Minimization Examples 10

End of Week 1: Module 5 Thank You Algebraic Minimization Examples 11