Karnaugh Maps

# Karnaugh Maps

## Karnaugh Maps

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##### Presentation Transcript

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’