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l7_15a_6

# l7_15a_6 - Last time 2-level simplification Goal Find a...

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1 ECE 15A Fundamentals of Logic Design Lecture 7 Malgorzata Marek-Sadowska Electrical and Computer Engineering Department UCSB 2 Last time: 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 3 Last time: Karnaugh map method ± K-map is an alternative method of representing the truth table that helps visualize adjacencies in up to 6 dimensions. ± For > 6 inputs CAD tools are needed. 2-variable map 0 1 a b 0 0 2 1 1 3 F(a,b,c) , order of variables: a,b,c 00 01 11 10 0 1 ab c 0 2 6 4 1 3 7 5 b=1 a=1 c=1 3-variable map 00 01 11 10 ab cd 00 0 4 12 8 01 1 5 13 9 11 3 7 15 11 10 2 6 14 10 a d b c Numbering scheme: 00,01,11,10 Gray code - only a single bit changes from code to next code word. F(a,b,c,d), variables order is a,b,c,d. 4 3-variable maps: examples = ) 7 , 6 , 2 , 0 ( m f Order: c,a,b c’b’ ac f=ac+c’b’ b 00 01 11 10 ab c 0 1 1 0 0 1 0 0 1 1 0 1 3 2 4 5 7 6 a c 0 1 3 2 4 5 7 6 00 01 11 10 ab c 0 1 0 0 1 0 0 1 1 1 a c b ab ac bc f=ab+ac+bc Order: c,a,b. = c’a’b’+ c’ab’+cab’+cab f=m3+m5+m6+m7 = c’ab+ca’b+ cab’+cab 5 3-variable maps: examples = ) 7 , 5 , 3 , 2 , 1 ( ) , , ( m z y x g = ) 6 , 5 , 4 , 3 , 1 ( ) , , ( m b a c f Order: c,a,b b 0 1 3 2 4 5 7 6 00 01 11 10 ab c 0 1 0 1 1 0 1 1 0 1 a c f= b’c+bc’+a’b=b’c+bc’+a’c Simplify a function g = x’z+x’y+xy’z+yz Order: x,y,z. 0 1 3 2 4 5 7 6 00 01 11 10 yz x 0 1 0 1 1 1 0 1 1 0 y x z g=z+x’y = c’a’b+c’ab +ca’b’+ca’b+cab’ = x’y’z+ x’yz’+x’yz+xy’z+xyz 6 2-level simplification 4-variable map m0 m1 m3 m2 m4 m5 m7 m6 m12 m13 m15 m14 m8 m9 m11 m10 1 square : minterm, 4 literals 2 adjacent squares: product term of 3 literals 4 adjacent squares: product term of 2 literals 8 adjacent squares: product term of 1 literal 16 adjacent squares: constant 1 Order: w,x,y,z z wx yz 00 01 11 10 00 01 11 10 x w y z y x w z y x w z y x w z y x w z y x w yz x w z wxy z xy w z y x w xyz w wxyz z y wx z y wx yz x w z y

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l7_15a_6 - Last time 2-level simplification Goal Find a...

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