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System Design

System Design. Lecture 4, 1 st year 2011/2012 Faculty of Information systems and Computer Science. Items. Four variable map. Examples. Product-of-Sums simplification Quiz no.1, 18/3/2012, on chapter 2. Four variable map. Y. YZ. 00 01 11 10. WX. 00 01 11 10.

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System Design

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  1. System Design Lecture 4, 1st year 2011/2012 Faculty of Information systems and Computer Science

  2. Items • Four variable map. • Examples. • Product-of-Sums simplification • Quiz no.1, 18/3/2012, on chapter 2.

  3. Four variable map. Y YZ 00 01 11 10 WX 00 01 11 10 X W Z

  4. Rules: • One square represents a minterm of four variables. • A rectangle of 2 squares represents a product term of three literals. • A rectangle of 4 squares represents a product term of two literals. • A rectangle of 8 squares represents a product term of one literal. • A rectangle of 16 squares represents a logic 1.

  5. Example: Y • Simplifying a 4-variable function with a map: • F(W,X,Y,Z)= ∑m(0,1,2,4,5,6,8,9,12,13,14) • F=Y*+W*Z*+XZ* YZ 00 01 11 10 WX 00 01 11 10 X W Z

  6. Example: C 00 01 11 10 • Simplifying a 4-variable function with a map • F=A*B*C*+B*CD*+AB*C*+A*BCD* • The term of 3 literals is represented by two squares. • =C*B*+CD*A*+D*B* AB CD 00 01 11 10 B A D

  7. Product-of-Sums simplification C CD 00 01 11 10 AB • Simplify the following boolean function in product-of-sums form: • F(A,B,C,D)=∑m(0,1,2,5,8,9,10) • Solution: • The 1’s marked in the map represent the minterm. • The squares marked with 0’s • Represent the minterm not included in F and therefore denote the complement of F. 00 01 11 10 B AA A D

  8. Continue • F*=AB+CD+BD* • Taking the dual and complement each literal: • F=(A*+B*)(C*+D*)(B*+D)

  9. Example: • Simplify the maxterm: • F= (A*+B*+C)(B+D) • Solution:1. Obtain its complement:F*=ABC*+B*D* • 2. Marks0’s in the squares representing the minterms of F*, the remaining squares are marked with1’s. • 3. Combining the 1’s gives the simplified expression in sum-of-products form. • 4. Combining 0’s and then complement gives the simplified expression in product-of-sums form.

  10. Continue C CD Sum of product: A*B+CD+CB+DB* 00 01 11 10 AB 00 01 11 10 B A D

  11. Examples: Y YZ WX 00 01 11 10 • Simplify the following Boolean Functions in product-of-sums form: • F(W,X,Y,Z)= ∑m(0,1,2,6,8,9,10,13) • F*=YZ+Y*XW*+WXZ* • Taking the dual and complementing • Each literal: • F=(Y*+Z*)(Y+X*+W)(W*+X*+Z) 00 01 11 10 X W Z

  12. Examples: C CD Simplify the following Boolean Functions in product-of-sums form: F(A,B,C,D)= Π M(1,3,5,6,7,9,10,11,14) F*=CD*+ABD+A*B*D* Taking the dual and complementing Each literal: F=(C*+D)(A*+B*+D*)(A+B+D) 00 01 11 10 AB 00 01 11 10 B A A D

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