TongLecture6 - CS281 Lecture 6 Tuanjie Tong So far we...

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CS281 Lecture 6 Tuanjie Tong
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So far we learned: Boolean functions Truth table Algebraic manipulation Canonical forms of Boolean function Standard forms of Boolean function Nonstandard forms Reduction of operators of functions Problem : don’t know if a standard or nonstandard form consists of minimum number of operators or not
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Next: Learn a procedure that allows us to find the nonstandard form that has minimum number of operators. Why is it interesting? Expression with fewer operators means fewer gates and fewer inputs per gate, and leads to less costly implementations.
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“Fewer” means “better”? True regarding implementation Not always true regarding performance (signal delay) Consider this example (with multiple inputs AND, OR gates): F=xy+xy’z+xy’w=x(y+y’z+y’w) =x(y+y’(z+w))
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Karnaugh Map (K-Map) Two-variable Map: Examples: AND, OR, XOR… x’y’ x’y xy’ xy x y 0 1 0 1 x y minterm 0 0 x’y’ 0 1 x’y 1 0 xy’ 1 1 xy
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K-Map (cntd…) M-subcube A rectangle containing adjacent 1’s Number of 1’s =2 m
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K-Map (cntd…) Choose as few subcubes as possible while still covering all 1-minterms. (Leading to less OR operators) Choose the largest possible subcubes (Leading to less AND operators) These two goals are consistent.
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K-Map (cntd…) Three variable K-Map – Example: c i+1 with x i , y i and c i in a full-adder x’y’z’ x’y’z x’yz x’yz’ xy’z’ xy’z xyz xyz’ x yz 00 01 11 10 0 1
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K-Map (cntd…) c i+1 =x i y i +c i x i +c i y i 1 1 1 1 c i x i y i 00 01 11 10 0 1
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K-Map (cntd…) 4-Variable K-Map m 0 m 1 m 3 m 2 m 4 m 5 m 7 m 6 m 12 m 13 m 15 m 14 m 8 m 9 m 11 m 10 xy zw 00 01 11 10 00 01 11 10
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K-Map (cntd…) 5-Variable K-Map
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TongLecture6 - CS281 Lecture 6 Tuanjie Tong So far we...

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