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Karnaugh Maps

100. 010. 111. 001. 101. 011. 110. 000. 00. 01. 11. 10. Karnaugh Maps. Five Variable Karnaugh Maps. Adjacent columns. abc. de. 24. 16. 0. 8. 28. 20. 4. 12. 25. 17. 1. 9. 29. 21. 5. 13. 27. 19. 3. 11. 31. 23. 7. 15. 26. 18. 2. 10. 30. 22. 6. 14.

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Karnaugh Maps

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  1. 100 010 111 001 101 011 110 000 00 01 11 10 Karnaugh Maps Five Variable Karnaugh Maps Adjacent columns abc de 24 16 0 8 28 20 4 12 25 17 1 9 29 21 5 13 27 19 3 11 31 23 7 15 26 18 2 10 30 22 6 14 4 variable K-map 4 variable K-map • Five Karnaugh Maps : “mirror” • 3 variables are laid out horizontally • 2 variables are laid out vertically

  2. 010 110 100 000 101 001 111 011 00 01 11 10 Karnaugh Maps Five Variable Karnaugh Maps Adjacent columns abc de 16 24 0 8 20 28 4 12 17 25 1 9 21 29 5 13 19 27 3 11 23 31 7 15 18 26 2 10 22 30 6 14 4 variable K-map 4 variable K-map • Five Karnaugh Maps : “stacked” • 3 variables are laid out horizontally • 2 variables are laid out vertically

  3. 010 100 001 011 110 111 101 000 00 01 11 10 Karnaugh Maps Simplification Using Five Variable Karnaugh Maps abc de 24 16 0 8 28 20 4 12 1 1 1 1 25 17 1 9 29 21 5 13 27 19 3 11 31 23 7 15 26 18 2 10 30 22 6 14 1 1 1 1 T = f (a,b,c,d,e) =  (0,2,8,10,16,18,24,26) • T = c’e’ : “mirror” • Vertical alignment produces logical adjacency • {0} : {16}, {2} : {18}, etc.

  4. 010 110 000 100 101 001 111 011 00 01 11 10 Karnaugh Maps Simplification using Five Variable Karnaugh Maps vwx yz 16 24 0 8 20 28 4 12 17 25 1 9 21 29 5 13 1 1 1 1 19 27 3 11 23 31 7 15 1 1 1 1 18 26 2 10 22 30 6 14 • R= f(v,w,x,y,z) = (5,7,13,15,21,23,29,31) • Stacked layout • 3 variable reduction : v,w,y • R= xz

  5. 010 110 100 000 101 001 111 011 00 01 11 10 Karnaugh Maps Simplification using Five Variable Karnaugh Maps abc de 16 24 0 8 20 28 4 12 1 1 1 1 17 25 1 9 21 29 5 13 1 1 1 1 19 27 3 11 23 31 7 15 1 1 1 1 18 26 2 10 22 30 6 14 1 1 1 1 • W= f(a,b,c,d,e) = (1,3,4.6.9.11.12.14.17.20,22,25,27,28,3023,29,31) • Two 3 variable reductions • a,b,d : ce’ • a,b,d : c’e • W= ce’+ce’

  6. 010 110 100 000 101 001 111 011 00 01 11 10 Karnaugh Maps Simplification using Five Variable Karnaugh Maps vwx yz 16 24 0 8 20 28 4 12 1 17 25 1 9 21 29 5 13 1 1 1 1 1 19 27 3 11 23 31 7 15 1 1 1 1 1 18 26 2 10 22 30 6 14 1 • J= f(v,w,x,y,z) = (5,7,13,15,21,23,29,31) • 2 EPI : • wz • v’w’x • J= wz+v’w’x

  7. 110 010 000 100 001 101 111 011 100 000 001 101 111 011 010 110 Karnaugh Maps Simplification Using Six Variable Karnaugh Maps abc def 32 48 0 16 40 56 8 24 33 49 1 17 41 57 9 25 35 51 3 19 43 59 11 27 34 50 2 18 42 58 10 26 36 52 4 20 44 60 12 28 37 53 5 21 45 61 13 29 39 55 23 47 63 15 31 7 38 54 6 22 46 62 14 30 • Stacking sequence ad(01)-> ad(11)-> ad(10) ad(00)->

  8. 010 110 100 000 001 101 011 111 100 000 001 101 111 011 110 010 Karnaugh Maps Simplification Using Six Variable Karnaugh Maps abc def 32 48 0 16 40 56 8 24 33 49 1 17 41 57 9 25 1 1 1 1 35 51 3 19 43 59 11 27 1 1 1 1 34 50 2 18 42 58 10 26 36 52 4 20 44 60 12 28 37 53 5 21 45 61 13 29 1 1 1 1 K=cf 39 55 23 47 63 15 31 7 1 1 1 1 38 54 6 22 46 62 14 30 • K= f(a,b,c,d,e,f)= (0,2,4,6,8,10,12,14,16,18,20,22,24,32,40,48,56

  9. 110 010 000 100 001 101 011 111 100 000 101 001 011 111 110 010 Karnaugh Maps Simplification Using Six Variable Karnaugh Maps abc def d’e’f’ 32 48 0 16 40 56 8 24 1 1 1 1 1 1 1 1 33 49 1 17 41 57 9 25 35 51 3 19 43 59 11 27 34 50 2 18 42 58 10 26 1 1 1 a’b’f’ 36 52 4 20 44 60 12 28 1 1 1 37 53 5 21 45 61 13 29 L= d’e’f’ +a’b’f’ +a’c’f’ 39 55 23 47 63 15 31 7 38 54 6 22 46 62 14 30 a’c’f’ 1 1 1 • L= f(a,b,c,d,e,f)= (0,2,4,6,8,10,12,14,16,18,20,22,24,32,40,48,56

  10. Karnaugh Maps Incompletely Specified Functions • Completely specified • Output value is known for every possible combination of input • Incompletely specified • Truth table does not generate an output value for every possible combination of input variables • Don’t Care term • Minterm or Maxterms that are not used as part of output function

  11. Karnaugh Maps Incompletely Specified Functions Don’t care terms • 1010, 1011,1100, 1101,1110, and 1111 Write eq. for output A,B,C,D • A=f(W,X,Y,Z) =(5,6,7,8,9) + d(10,11,12,13,14,15) • B=f(W,X,Y,Z) =(1,2,3,4,9) + d(10,11,12,13,14,15) • C=f(W,X,Y,Z) =(0,3,4,7,8) + d(10,11,12,13,14,15) • D=f(W,X,Y,Z) =(0,2,4,6,8) + d(10,11,12,13,14,15)

  12. Don’t Care Terms • Develop the truth table that describes the input/output relationship • Determine if all of the input combinations are used to generate output(s) • If so, then no don’t care terms exist • If not, then those combinations of input variables not used to determine output values are don’t care terms • Once the don’t care terms have been identified, use a separate symbol, in the K-map squares, so they will not be confused with normal Minterms or Maxterms input variables never occurs • Create as large an EPI grouping as possible, including don’t care terms that have been combined with normal Minterms • Do not group don’t care term by themselves Procedure

  13. Karnaugh Maps Don’t Care Terms • Don’t care terms are the same for each output variable in the problem, because the same set of input combinations are used • Don’t care terms are distinguished from regular minterms in that it does not matter whether we assign “0” or “1” • These combinations of input variables never occurs • Distinct advantage when simplifying the output eq.

  14. Don’t Care Terms A=W+XY+XZ B=X’Y+X’Z+XY’Z’ C=Y’Z’+YZ D=Z’

  15. Karnaugh Maps BCD to EX-3 Code Conversion Circuits A=W+XY+XZ B=X’Y+X’Z+XY’Z’ C=Y’Z’+YZ D=Z’

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