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Overview

Overview. Last lecture Correspondence between canonical forms Boolean cubes Karnaugh maps Today More Karnaugh map examples Don’t cares in Karnaugh maps Design examples. A. 000 010 001 011. 110 100 111 101. C. B. Adjacencies in Karnaugh maps. Wrap from first to last column

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Overview

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  1. Overview • Last lecture • Correspondence between canonical forms • Boolean cubes • Karnaugh maps • Today • More Karnaugh map examples • Don’t cares in Karnaugh maps • Design examples CSE 370 – Winter 2002 – Logic minimization - 1

  2. A 000 010 001 011 110 100 111 101 C B Adjacencies in Karnaugh maps • Wrap from first to last column • Wrap top row to bottom row 011 111 110 010 001 B 101 C 100 000 A CSE 370 – Winter 2002 – Logic minimization - 2

  3. B’ A 1 1 0 0 AB + ACin + BCin A B 0 0 0 1 1 0 1 1 Cin A 1 0 0 0 0 1 1 1 B C AC + B’C’ + AB’ B Karnaugh map examples • F = • Cout = • f(A,B,C) = m(0,4,6,7) obtain thecomplementof the function by covering 0’srather than 1’s CSE 370 – Winter 2002 – Logic minimization - 3

  4. A A A A 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 1 1 1 1 0 0 0 C C C = AC + B’C’ B B B = BC’ + A’C More Karnaugh map examples G(A,B,C) = F(A,B,C) = m(0,4,5,7) F' simply replace 1's with 0's and vice versa F'(A,B,C) =  m(1,2,3,6) CSE 370 – Winter 2002 – Logic minimization - 4

  5. C + A’BD + B’D’ 1111 0111 0 1 0 0 1 0 0 1 Y Z W 1000 1 1 1 1 1 1 1 1 0000 X C Karnaugh map: 4-variable example • F(A,B,C,D) = m(0,2,3,5,6,7,8,10,11,14,15)F = A D B find the smallest number of the largest possible subcubes to cover the ON-set (fewer terms with fewer inputs per term) CSE 370 – Winter 2002 – Logic minimization - 5

  6. A’D + B’C’D A X 0 X 1 0 0 1 1 D 0 0 0 0 1 1 0 X C B Karnaugh maps: don’t cares • f(A,B,C,D) = m(1,3,5,7,9) + d(6,12,13) • without don't cares • f = CSE 370 – Winter 2002 – Logic minimization - 6

  7. + C'D A'D by using don't care as a "1"a 2-cube can be formed rather than a 1-cube to coverthis node A X 0 X 1 0 0 1 1 D 0 0 0 0 1 1 0 X C B Karnaugh maps: don’t cares (cont’d) • f(A,B,C,D) = m(1,3,5,7,9) + d(6,12,13) • f = A'D + B'C'D without don't cares • f = with don't cares don't cares can be treated as1s or 0sdepending on which is more advantageous CSE 370 – Winter 2002 – Logic minimization - 7

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