ece15_7_2011_6

ece15_7_2011_6 - Last time: Karnaugh map method K-map is an...

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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: 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 2 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. 3 4-adjacent squares in 3-variable map 0 1 2 4 5 7 6 x’y’z’ x’y’z x’yz x’yz’ xy’z’ xy’z xyz xyz’ x yz 00 01 11 10 0 1 z y Order:x,y,z f= m0 + m1+ m4 + m5= x’y’z’+ x’y’z + xy’z’ + xy’z = x’y’(z’ + z) + xy’(z’+z)= x’y’ + xy’ = (x’ + x)y’ = y’ 0 1 2 4 5 7 6 x’y’z’ x’y’z x’yz x’yz’ xy’z’ xy’z xyz xyz’ x yz 00 01 11 10 0 z y g = m0 + m2 + m4 + m6 = x’y’z’ + x’yz’ + xy’z’ + xyz’ = x’z’ (y’ + y) + xz’ (y’ + y) = z’(x’+x) = z’ 1 square : minterm of 3 literals 2 adjacent squares: product term of 2 literals 4 adjacent squares: product term of 1 literal 8 adjacent squares: constant 1 4 4-adjacent squares in 3-variable map 0 2 4 7 6 x’y’z’ x’y’z x’yz x’yz’ xy’z’ xy’z xyz xyz’ x yz 00 01 11 10 0 1 z y 0 2 4 7 6 x’y’z’ x’y’z x’yz x’yz’ xy’z’ xy’z xyz xyz’ x yz 00 01 11 10 0 z y 0 1 2 4 5 7 6 x’y’z’ x’y’z x’yz x’yz’ xy’z’ xy’z xyz xyz’ x yz 00 01 11 10 0 1 z y These adjacencies are not useful in simplification. 5 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 x w z y x w wx yz 6 2-level simplification yz x w z y x w xyz w z xy w wxyz z wxy yz x w z y x w + + + + + + + = y x w (z+z’)+ xy w (z+z’)+ y x w
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This note was uploaded on 02/26/2011 for the course ECE 15A taught by Professor M during the Winter '08 term at UCSB.

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ece15_7_2011_6 - Last time: Karnaugh map method K-map is an...

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