Karnaugh Maps
15
22年 3年 19年 1 Karnaugh Maps Karnaugh Maps Five Variable Karnaugh Maps Five Variable Karnaugh Maps 0 4 12 12 8 00 00 01 01 11 11 10 10 000 000 001 001 011 011 010 010 abc abc de de 1 5 13 13 9 3 7 15 15 11 11 2 6 14 14 10 10 Five Karnaugh Maps : Five Karnaugh Maps : “mirror” “mirror” • 3 variables 3 variables are laid out horizontally are laid out horizontally • 2 variables 2 variables are laid out vertically are laid out vertically 24 24 28 28 20 20 16 16 110 110 111 111 101 101 100 100 25 25 29 29 21 21 17 17 27 27 31 31 23 23 19 19 26 26 30 30 22 22 18 18 4 variable K-map 4 variable K-map Adjacent columns Adjacent columns
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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. - PowerPoint PPT Presentation
Transcript of Karnaugh Maps
23年 4月 20日1
Karnaugh MapsKarnaugh MapsFive Variable Karnaugh MapsFive Variable Karnaugh Maps
00 44 1212 880000
0101
1111
1010
000000 001001 011011 010010abcabc
dede
11 55 1313 99
33 77 1515 1111
22 66 1414 1010
Five Karnaugh Maps : Five Karnaugh Maps : “mirror”“mirror”• 3 variables 3 variables are laid out horizontallyare laid out horizontally• 2 variables2 variables are laid out vertically are laid out vertically
2424 2828 2020 1616110110 111111 101101 100100
2525 2929 2121 1717
2727 3131 2323 1919
2626 3030 2222 1818
4 variable K-map 4 variable K-map
Adjacent columnsAdjacent columns
23年 4月 20日2
Karnaugh MapsKarnaugh MapsFive Variable Karnaugh MapsFive Variable Karnaugh Maps
00 44 1212 880000
0101
1111
1010
000000 001001 011011 010010abcabc
dede
11 55 1313 99
33 77 1515 1111
22 66 1414 1010
1616 2020 2828 2424100100 101101 111111 110110
1717 2121 2929 2525
1919 2323 3131 2727
1818 2222 3030 2626
4 variable K-map4 variable K-map
Adjacent columnsAdjacent columns
Five Karnaugh Maps : Five Karnaugh Maps : “stacked”“stacked”
3 variables 3 variables are laid out horizontallyare laid out horizontally 2 variables2 variables are laid out vertically are laid out vertically
23年 4月 20日3
Karnaugh MapsKarnaugh MapsSimplification Using Five Variable Karnaugh MapsSimplification Using Five Variable Karnaugh Maps
T = c’e’ : T = c’e’ : “mirror”“mirror” Vertical alignment produces logical adjacencyVertical alignment produces logical adjacency
• {0} : {16}, {2} : {18}, etc.{0} : {16}, {2} : {18}, etc.
00 44 1212 88
0000
0101
1111
1010
000000 000101 001111 001010abcabc
dede
11 55 1313 99
33 77 1515 1111
22 66 1414 1010
2424 2828 2020 1616111010 111111 110101 110000
2525 2929 2121 1717
2727 3131 2323 1919
2626 3030 2222 1818
11
11 11
11 11 11
1111
T = T = f f (a,b,c,d,e) = (a,b,c,d,e) = (0,2,8,10,16,18,24,26) (0,2,8,10,16,18,24,26)
23年 4月 20日4
Karnaugh MapsKarnaugh MapsSimplification using Five Variable Karnaugh MapsSimplification using Five Variable Karnaugh Maps
R= f(v,w,x,y,z) = R= f(v,w,x,y,z) = (5,7,13,15,21,23,29,31)(5,7,13,15,21,23,29,31) Stacked layoutStacked layout
• 3 variable reduction : 3 variable reduction : vv ,,w,yw,y• R= xzR= xz
00 44 1212 880000
0101
1111
1010
000000 001001 011011 010010vwxvwx
yzyz
11 55 1313 99
33 77 1515 1111
22 66 1414 1010
1616 2020 2828 2424100100 101101 111111 110110
1717 2121 2929 2525
1919 2323 3131 2727
1818 2222 3030 2626
11 11
11 11
11
1111
11
23年 4月 20日5
Karnaugh MapsKarnaugh MapsSimplification using Five Variable Karnaugh MapsSimplification using Five Variable Karnaugh Maps
W= f(a,b,c,d,e) = W= f(a,b,c,d,e) = ((1,3,4.6.9.11.12.14.17.20,22,25,27,28,3023,29,311,3,4.6.9.11.12.14.17.20,22,25,27,28,3023,29,31))• Two 3 variable reductionsTwo 3 variable reductions
o a,b,d : ce’a,b,d : ce’o a,b,d : c’ea,b,d : c’e
• W= ce’+ce’W= ce’+ce’
00 44 1212 880000
0101
1111
1010
000000 001001 011011 010010abcabc
dede
11 55 1313 99
33 77 1515 1111
22 66 1414 1010
1616 2020 2828 2424100100 101101 111111 110110
1717 2121 2929 2525
1919 2323 3131 2727
1818 2222 3030 2626
11 11
11 11
11
1111
11
1111 1111
11111111
23年 4月 20日6
Karnaugh Karnaugh MapsMapsSimplification using Five Variable Karnaugh MapsSimplification using Five Variable Karnaugh Maps
J= f(v,w,x,y,z) = J= f(v,w,x,y,z) = (5,7,13,15,21,23,29,31)(5,7,13,15,21,23,29,31)• 2 EPI :2 EPI :
• wzwz• v’w’xv’w’x
• J= wz+v’w’xJ= wz+v’w’x
00 44 1212 880000
0101
1111
1010
000000 001001 011011 010010vwxvwx
yzyz
11 55 1313 99
33 77 1515 1111
22 66 1414 1010
1616 2020 2828 2424100100 101101 111111 110110
1717 2121 2929 2525
1919 2323 3131 2727
1818 2222 3030 2626
11 11
11 11
11
1111
1111
11
11
11
23年 4月 20日7
Karnaugh Karnaugh MapsMaps
00 88 2424 1616000000
001001
011011
010010
000000 001001 011011 010010abcabc
defdef
11 99 2525 1717
33 1111 2727 1919
22 1010 2626 1818
3232 4040 5656 4848
110000 101101 111111 110110
3333 4141 5757 4949
3535 4343 5959 5151
3434 4242 5858 5050
44 1212 2828 2020
55 1313 2929 2121
77 1515 3131 2323
66 1414 3030 2222
3636 4444 6060 5252
3737 4545 6161 5353
3939 4747 6363 5555
3838 4646 6262 5454
110000
101101
111111
110110
Stacking sequenceStacking sequence
Simplification Using Six Variable Karnaugh MapsSimplification Using Six Variable Karnaugh Maps
ad(00)->ad(00)->ad(01)ad(01)->->ad(11)ad(11)->->ad(10)ad(10)
23年 4月 20日8
Karnaugh MapsKarnaugh Maps
00 88 2424 1616000000
001001
011011
010010
000000 001001 011011 010010abcabc
defdef
11 99 2525 1717
33 1111 2727 1919
22 1010 2626 1818
3232 4040 5656 4848
100100 101101 111111 110110
3333 4141 5757 4949
3535 4343 5959 5151
3434 4242 5858 5050
44 1212 2828 2020
55 1313 2929 2121
77 1515 3131 2323
66 1414 3030 2222
3636 4444 6060 5252
3737 4545 6161 5353
3939 4747 6363 5555
3838 4646 6262 5454
100100
101101
111111
110110
K= f(a,b,c,d,e,f)= (0,2,4,6,8,10,12,14,16,18,20,22,24,32,40,48,56K= f(a,b,c,d,e,f)= (0,2,4,6,8,10,12,14,16,18,20,22,24,32,40,48,56
Simplification Using Six Variable Karnaugh MapsSimplification Using Six Variable Karnaugh Maps
1 1
1 1
1 1
1 1
1 1
1 1
1 1
1 1
K=cfK=cf
23年 4月 20日9
Karnaugh MapsKarnaugh Maps
00 88 2424 1616000000
001001
011011
010010
000000 001001 011011 010010abcabc
defdef
11 99 2525 1717
33 1111 2727 1919
22 1010 2626 1818
3232 4040 5656 4848
100100 101101 111111 110110
3333 4141 5757 4949
3535 4343 5959 5151
3434 4242 5858 5050
44 1212 2828 2020
55 1313 2929 2121
77 1515 3131 2323
66 1414 3030 2222
3636 4444 6060 5252
3737 4545 6161 5353
3939 4747 6363 5555
3838 4646 6262 5454
100100
101101
111111
110110
L= f(a,b,c,d,e,f)= (0,2,4,6,8,10,12,14,16,18,20,22,24,32,40,48,56L= f(a,b,c,d,e,f)= (0,2,4,6,8,10,12,14,16,18,20,22,24,32,40,48,56
Simplification Using Six Variable Karnaugh MapsSimplification Using Six Variable Karnaugh Maps
1 11 1
1 1
1 1
1
1 11 1 d’e’f’d’e’f’
1
1 1
1a’b’f’a’b’f’
a’c’f’a’c’f’
L= d’e’f’d’e’f’ +a’b’f’a’b’f’ +a’c’f’a’c’f’
23年 4月 20日10
Karnaugh Karnaugh MapsMapsIncompletely Specified FunctionsIncompletely Specified Functions
Completely specifiedCompletely specified Output value is known for every possible combination of Output value is known for every possible combination of inputinput
Incompletely specifiedIncompletely specified
Truth table does not generate an output value for every Truth table does not generate an output value for every possible combination of input variablespossible combination of input variables
Don’t Care termDon’t Care term
Minterm or Maxterms that are not used as part of Minterm or Maxterms that are not used as part of output output functionfunction
23年 4月 20日11
Karnaugh Karnaugh MapsMapsIncompletely Specified FunctionsIncompletely Specified Functions
Don’t care termsDon’t care terms
1010, 1011,1100, 1101,1110, and 1010, 1011,1100, 1101,1110, and 11111111
Write eq. for output A,B,C,DWrite eq. for output A,B,C,D
A=f(W,X,Y,Z)A=f(W,X,Y,Z) ==(5,6,7,8,9)(5,6,7,8,9) + +
d(10,11,12,13,14,15)d(10,11,12,13,14,15) B=f(W,X,Y,Z)B=f(W,X,Y,Z)
==(1,2,3,4,9)(1,2,3,4,9) + + d(10,11,12,13,14,15)d(10,11,12,13,14,15)
C=f(W,X,Y,Z)C=f(W,X,Y,Z) ==(0,3,4,7,8)(0,3,4,7,8) + +
d(10,11,12,13,14,15)d(10,11,12,13,14,15) D=f(W,X,Y,Z)D=f(W,X,Y,Z)
==(0,2,4,6,8)(0,2,4,6,8) + + d(10,11,12,13,14,15)d(10,11,12,13,14,15)
23年 4月 20日12
Don’t Care Don’t Care TermsTermsProcedureProcedure
Develop the truth table that describes the Develop the truth table that describes the input/output relationshipinput/output relationship
Determine if all of the input combinations are used to Determine if all of the input combinations are used to generate output(s)generate output(s) If so, then If so, then no don’t care terms existno don’t care terms exist If not, then If not, then those combinations of input variables not usedthose combinations of input variables not used
to determine output values are to determine output values are don’t care termsdon’t care terms
Once the don’t care terms have been identified, use a Once the don’t care terms have been identified, use a separate symbol, in the K-map squares, so they will not separate symbol, in the K-map squares, so they will not be confused with normal Minterms or Maxterms input be confused with normal Minterms or Maxterms input variables never occursvariables never occurs
Create as large an EPI grouping as possible, including Create as large an EPI grouping as possible, including don’t care terms that have been combined with normal don’t care terms that have been combined with normal MintermsMinterms
Do not group don’t care term by themselvesDo not group don’t care term by themselves
23年 4月 20日13
Karnaugh Karnaugh MapsMapsDon’t Care TermsDon’t Care Terms
Don’t care terms are the same for each output Don’t care terms are the same for each output variable in the problem, because the same set variable in the problem, because the same set of input combinations are usedof input combinations are used
Don’t care terms are distinguished from Don’t care terms are distinguished from regular minterms in that it does not matter regular minterms in that it does not matter whether we assign “0” or “1”whether we assign “0” or “1”
These combinations of input variables never These combinations of input variables never occursoccurs
Distinct advantage when simplifying the output Distinct advantage when simplifying the output eq.eq.
23年 4月 20日14
Don’t Care TermsDon’t Care Terms
A=W+XY+XZA=W+XY+XZ B=X’Y+X’Z+XY’Z’B=X’Y+X’Z+XY’Z’
C=Y’Z’+YZC=Y’Z’+YZ D=Z’D=Z’
23年 4月 20日15
Karnaugh Karnaugh MapsMapsBCD to EX-3 Code Conversion CircuitsBCD to EX-3 Code Conversion Circuits
A=W+XY+XZA=W+XY+XZ
B=X’Y+X’Z+XY’Z’B=X’Y+X’Z+XY’Z’
C=Y’Z’+YZC=Y’Z’+YZ
D=Z’D=Z’
