Digital Circuits and Systems Algebraic Minimization Examples - - PowerPoint PPT Presentation

Shankar Balachandran* Associate Professor, CSE Department Indian Institute of Technology Madras

*Currently a Visiting Professor at IIT Bombay

Digital Circuits and Systems

Spring 2015 Week 1 Module 5

Algebraic Minimization Examples

Summary of Minterms and Maxterms

Algebraic Minimization Examples 2

Algebraic Minimization Examples 3

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,……,2n –1}. It can be shown that the minterm indices and maxterm indices are complementary.

 Example: Assume a 3 variable expression F(x,y,z).

i i

m M 

That is,

   

6 5 3 2 7 4 1

6 5 3 2 7 4 1 M M M M M m m m , , , , , ,          

Algebraic Minimization Examples 4

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 5

Example:

 Simplify to SOP form:

 

   y

z z y x z , y , x F   

 

y z y z y x    y z y z x y y x    1     y z y z x y x

 

1    x y z y x 1    y z y x y z y x  

 

y z y x z , y , x F  

Algebraic Minimization Examples 6

Example:

 Write the following to canonical SOP (sum of minterms) form.

 

y z y x z , y , x f  

   

z y x z y x z y x z y x x x y z z z y x         z y x z y x z y x z y x z y x z y x      

   

  6 3 2 , , z , y , x f

Algebraic Minimization Examples 7

Example:

 Simplify to POS form:

 

z y x y x z y x z , y , x f   

   

y y y x x y x y x y x z z y x           1

Algebraic Minimization Examples 8

Example:

 Simplify to SOP and POS forms.

 



d c b c b a   d c c c b d c b a b a    

 

 

SOP form POS form ab abcd bc ab cd bc ab bc ...... b a c ......            1

Algebraic Minimization Examples 9

Example:

 Simplify to POS and expand to canonical POS (product of maxterms).

 

z x y x z , y , x f  

 



z y x x y x   

  

 

 

 

POS form x x y x x z y z x y x z y z ........         

   

z y x x z y y x z z y x       

  









 

  

z y x z y x z y x z y x z y x z y x z y x z y x z y x z y x                      

   

Canonical POS form f x,y,z , , , ......   0 2 4 5

Algebraic Minimization Examples 10

Example:

 Simplify to SOP form:

 

    wxz z y x w

 

   

   

z x w z y x w z x w z y x w       

 

 

wxz z y x w   

          

 

z y x w x w z y z z x w z x w z y z y x z y w z x w z x w z y y y x y w x w z x w z y x y w                      1

End of Week 1: Module 5

Thank You

Algebraic Minimization Examples 11