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Unformatted text preview: 1 ECE 15A Fundamentals of Logic Design Lecture 6 Malgorzata Marek-Sadowska Electrical and Computer Engineering Department UCSB 2 Functionally complete operations We know that (OR, AND,NOT) are functionally complete. (OR,NOT) is also functionally complete (AB) = (AB)=((AB))=(A+B) B A A B F F 3 Functionally complete operations (NOR) is also functionally complete (AB) = (AB)=((AB))=(A+B) (A+A) = A B A A B F F A B A A A A A B B A F F F F A B 4 Example 2 level realization f(a,b,c)=abc+abc+abc+abc+abc a b c f Direct representation. 9 gates, 11 nets, 15 literals. 5 Simplification f b a c 2 gates, 4 nets, 3 literals f=abc+abc+abc+abc+abc = ab(c+c) + abc+abc+abc+abc = ab(c+c) + ab(c+c)+ ac(b+b) = ab+ab+ac = b(a+a)+ac = b+ac 6 2-level simplification Goal: Find a representation of f which is composed of as few product terms as possible. Minimize the number of literals. Example ac b f abc c ab c b a bc a c b a f + = + + + + = min f has 5 product terms, 15 literals min f has 2 product terms, 3 literals 2 7 Example a b c f f 0 0 0 a + b + c = M0 1 0 0 1 a + b + c = M1 1 0 1 0 a + b+ c = M2 1 0 1 1 a + b+ c = M3 1 1 0 0 a+ b + c = M4 1 1 0 1 a+ b + c = M5 1 1 1 0 a+ b+ c = M6 1 1 1 1 a+ b+ c = M7 1 ) )( )( ( ) 2 , 1 , ( ) , , ( c b a c b a c b a M c b a f + + + + + + = = (Canonical product of sums). 8 Circuit realization Canonical product of sums a b c 6 gates, 8 nets, 9 literals 9 Example ' ' ' ' ) ' ( ' ' ) ' ( ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' 2 1 ) , , ( ' c a b a b b c a c c b a c b a bc a c b a c b a bc a c b a c b a m m m c b a f + =...
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- Winter '08