chapter 3_logic circuits
DESCRIPTION
Logic circuits courseTRANSCRIPT
-
CE211 Digital SystemsCE211 Digital SystemsGate Level Minimization
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KarnaughKarnaugh
MapMap
Adjacent Squares
Number of squares = number of combinations
Each square represents a minterm
2 Variables 4 squares
3 Variables 8 squares
4 Variables 16 squares
Each two adjacent squares differ in one variable
Two adjacent minterms
can be combined together
Example: F = x y + x y
= x ( y + y )
= x
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TwoTwo--variable Mapvariable Map
m0 m1
m2 m3
x y Minterm
0 0 0 m01 0 1 m12 1 0 m23 1 1 m3
yxyxyxyx
yx 0 1
0
1
yx yx
yx yx
Note: adjacent squares horizontally and vertically NOT diagonally
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TwoTwo--variable Mapvariable Map
Examplex y F Minterm
0 0 0 0 m01 0 1 0 m12 1 0 0 m23 1 1 1 m3
yx 0 1
0
1
m0 m1
m2 m3
y
0 0
x 0 1
yxyxyxyx
yx yx
yx yx
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TwoTwo--variable Mapvariable Map
Examplex y F Minterm
0 0 0 0 m01 0 1 1 m12 1 0 1 m23 1 1 1 m3
yx 0 1
0
1
m0 m1
m2 m3
y
0 1
x 1 1
yxyxyxF ++=
yxx )( + )( yyx +xyF +=
yxyxyxyx
yx yx
yx yx
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ThreeThree--variable Mapvariable Map
m0 m1 m3 m2
m4 m5 m7 m6
x y z Minterm
0 0 0 0 m01 0 0 1 m12 0 1 0 m23 0 1 1 m34 1 0 0 m45 1 0 1 m56 1 1 0 m67 1 1 1 m7
zyx
y zx 00 01 11 10
0
1
zyxzyxzyxzyxzyxzyxzyx
zyx zyx zyxzyx
zyx zyx zyxzyx
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ThreeThree--variable Mapvariable Map
m0 m1 m3 m2m4 m5 m7 m6x y z F Minterm
0 0 0 0 0 m01 0 0 1 0 m12 0 1 0 1 m23 0 1 1 1 m34 1 0 0 1 m45 1 0 1 1 m56 1 1 0 0 m67 1 1 1 0 m7
y zx 00 01 11 10
0
1
Example
y
0 0 1 1
x 1 1 0 0z
=F yxyx +
zyxzyxzyxzyxzyxzyxzyxzyx
zyx zyx zyxzyxzyx zyx zyxzyx
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77 / / 143143yx
ThreeThree--variable Mapvariable Map
m0 m1 m3 m2m4 m5 m7 m6x y z F Minterm
0 0 0 0 0 m01 0 0 1 0 m12 0 1 0 0 m23 0 1 1 1 m34 1 0 0 1 m45 1 0 1 0 m56 1 1 0 1 m67 1 1 1 1 m7
y zx 00 01 11 10
0
1
Example
y
0 0 1 0
x 1 0 1 1z
=F zyzx + +Extra
zyxzyxzyxzyxzyxzyxzyxzyx
zyx zyx zyxzyxzyx zyx zyxzyx
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ThreeThree--variable Mapvariable Map
x y z F Minterm
0 0 0 0 0 m01 0 0 1 1 m12 0 1 0 0 m23 0 1 1 1 m34 1 0 0 0 m45 1 0 1 1 m56 1 1 0 0 m67 1 1 1 1 m7
Example y0 1 1 0
x 0 1 1 0z
zx zx
zyxzyxzyxzyxF +++=
)( yyzx + )( yyzx +
y
0 1 1 0
x 0 1 1 0z
zyxzyxzyxzyxzyxzyxzyxzyx
z
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ThreeThree--variable Mapvariable Map
m0 m1 m3 m2m4 m5 m7 m6x y z F Minterm
0 0 0 0 1 m01 0 0 1 0 m12 0 1 0 1 m23 0 1 1 0 m34 1 0 0 1 m45 1 0 1 1 m56 1 1 0 1 m67 1 1 1 0 m7
y zx 00 01 11 10
0
1
Example
y
1 0 0 1
x 1 1 0 1z
=F z yx+
zyxzyxzyxzyxzyxzyxzyxzyx
zyx zyx zyxzyxzyx zyx zyxzyx
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FourFour--variable Mapvariable Map
m0 m1 m3 m2m4 m5 m7 m6m12 m13 m15 m14m8 m9 m11 m10
w x y z Minterm0 0 0 0 0 m01 0 0 0 1 m12 0 0 1 0 m23 0 0 1 1 m34 0 1 0 0 m45 0 1 0 1 m56 0 1 1 0 m67 0 1 1 1 m78 1 0 0 0 m89 1 0 0 1 m9
10 1 0 1 0 m1011 1 0 1 1 m1112 1 1 0 0 m1213 1 1 0 1 m1314 1 1 1 0 m1415 1 1 1 1 m15
y zwx 00 01 11 10
00
01
11
10
zyxwzyxwzyxwzyxwzyxwzyxwzyxwzyxwzyxwzyxwzyxwzyxwzyxwzyxwzyxwzyxw
zyxw zyxw yzxw zyxw
zyxw zyxw xyzw zxyw
zyxw zyxw yzxw zyxw
zywx zywx wxyz zwxy
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FourFour--variable Mapvariable Map
w x y z F Minterm0 0 0 0 0 1 m01 0 0 0 1 1 m12 0 0 1 0 1 m23 0 0 1 1 0 m34 0 1 0 0 1 m45 0 1 0 1 1 m56 0 1 1 0 1 m67 0 1 1 1 0 m78 1 0 0 0 1 m89 1 0 0 1 1 m910 1 0 1 0 0 m1011 1 0 1 1 0 m1112 1 1 0 0 1 m1213 1 1 0 1 1 m1314 1 1 1 0 1 m1415 1 1 1 1 0 m15
y zwx 00 01 11 10
00011110
zyxwzyxwzyxwzyxwzyxwzyxwzyxwzyxwzyxwzyxwzyxwzyxwzyxwzyxwzyxwzyxw
zyxw zyxw yzxw zyxwzyxw zyxw xyzw zxyw
zyxw zyxw yzxw zyxwzywx zywx wxyz zwxy
Example
y1 1 0 11 1 0 1
xw
1 1 0 11 1 0 0
z
=F y zw+ + zx
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FourFour--variable Mapvariable Map
ExampleSimplify: F = A B C + B C D + A B C D + A B C
C
BA
D
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FourFour--variable Mapvariable Map
ExampleSimplify: F = A B C + B C D + A B C D + A B C
C
BA
D
1 1
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FourFour--variable Mapvariable Map
ExampleSimplify: F = A B C + B C D + A B C D + A B C
C
BA
D
1
1
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FourFour--variable Mapvariable Map
ExampleSimplify: F = A B C + B C D + A B C D + A B C
C
BA
D
1
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FourFour--variable Mapvariable Map
ExampleSimplify: F = A B C + B C D + A B C D + A B C
C
BA
D1 1
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FourFour--variable Mapvariable Map
ExampleSimplify: F = A B C + B C D + A B C D + A B C
C1 1 1
1B
A1 1 1
D
=F DB CB+ + DCA
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FiveFive--variable Mapvariable Map
DE DBC 00 01 11 10
00 m0 m1 m3 m2
01 m4 m5 m7 m6C
B11 m12 m13 m15 m14
10 m8 m9 m11 m10
E
A = 0
DE DBC 00 01 11 10
00 m16 m17 m19 m18
01 m20 m21 m23 m22C
B11 m28 m29 m31 m30
10 m24 m25 m27 m26
E
A = 1
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FiveFive--variable Mapvariable Map
A = 0
A = 1
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ImplicantsImplicants
C
1
1 1 1B
A1 1 1
1
D
Implicant:Gives F = 1
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Prime Prime ImplicantsImplicants
C
1
1 1 1B
A1 1 1
1
D
Prime Implicant:Cant grow beyond this size
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Essential Prime Essential Prime ImplicantsImplicants
C
1
1 1 1B
A1 1 1
1
D
Essential Prime Implicant:No other choice
Not essential
8 Implicants5 Prime
implicants
4 Essential
prime
implicants
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Product of Sums SimplificationProduct of Sums Simplification
w x y z F F0 0 0 0 0 1 01 0 0 0 1 1 02 0 0 1 0 1 03 0 0 1 1 0 14 0 1 0 0 1 05 0 1 0 1 1 06 0 1 1 0 1 07 0 1 1 1 0 18 1 0 0 0 1 09 1 0 0 1 1 010 1 0 1 0 0 111 1 0 1 1 0 112 1 1 0 0 1 013 1 1 0 1 1 014 1 1 1 0 1 015 1 1 1 1 0 1
y1 1 0 11 1 0 1
xw
1 1 0 11 1 0 0
z =Fy zw+ + zx
y1 1 0 11 1 0 1
xw
1 1 0 11 1 0 0
z
=F zy + yxw
yxwzyF +=
)()( yxwzyF +++=
F
ywz
zx
y
wzxy
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DonDontt--Care ConditionCare Condition
Example
otherwiseC
0depositedisnicleaif1{=
otherwiseB
0depositedisdimeaif1{=
otherwiseA
0depositedisquarteraif1{=
A B C $ Value0 0 0 $ 0.000 0 1 $ 0.050 1 0 $ 0.100 1 1 Not possible1 0 0 $ 0.251 0 1 Not possible1 1 0 Not possible1 1 1 Not possible
You can only drop one coin at
a time.
Used as dont care
http://www.vending101.com/snacks.htm -
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DonDontt--Care ConditionCare Condition
Example
A B C F0 0 0 00 0 1 10 1 0 10 1 1 x1 0 0 11 0 1 x1 1 0 x1 1 1 x
F
Dont care what value F may take
Logic Circuit
=
=
)7,6,5,3(),,(
)4,2,1(),,(
CBAd
CBAF
A
B
C
http://www.vending101.com/snacks.htm -
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DonDontt--Care ConditionCare Condition
Example
F
B
0 1 x 1
A 1 x x xC
BCAFCBACBACBAF
++=++=
A
B
C
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DonDontt--Care ConditionCare Condition
Example
y
x 1 1 x
x 1x
w1
1z
F (w, x, y, z) = (1, 3, 7, 11, 15)
d (w, x, y, z) = (0, 2, 5)
zwzyF +=
x = 0
x = 1
y
x x
0 x 0x
w0 0 0
0 0 0z
ywzF +=
x = 0x = 1
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QuineQuineMcCluskey Tabular Method MotivationMcCluskey Tabular Method Motivation
Karnaugh maps are effective for the minimizationof expressions with up to 5 or 6 inputs
difficult to use and error prone for circuits with many inputs.
Karnaugh maps depend on our ability to visually identify prime implicants and select a set of prime implicants that cover all minterms.
They do not provide a direct algorithm to be implemented in a computer.
For larger systems, we need a programmable method!!
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QuineQuine--McCluskeyMcCluskey
Quine, Willard, A way to simplify truth functions.American Mathematical Monthly, vol. 62, 1955.
Quine, Willard, The problem of simplifying truth functions.American Mathematical Monthly, vol. 59, 1952.
Willard van Orman Quine 1908-2000, Edgar Pierce Chair of Philosophy at Harvard University.http://members.aol.com/drquine/wv-quine.html
McCluskey Jr., Edward J. Minimization of Boolean Functions.Bell Systems Technical Journal, vol. 35, pp. 1417-1444, 1956
Edward J. McCluskey, Professor of ElectricalEngineering and Computer Science at Stanfordhttp://www-crc.stanford.edu/users/ejm/McCluskey_Edward.html
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Outline of the QuineOutline of the Quine--McCluskey MethodMcCluskey Method
1. Produce a minterm expansion (standard sum-of-products form) for a function F
2. Eliminate as many literals as possible bysystematically applying XY + XY
= X.
3. Use a prime implicant chart to select aminimum set of prime implicants
that
when ORed together produce F, and thatcontains a minimum number of literals.
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Determination of Prime ImplicantsDetermination of Prime Implicants
ABCD
+ ABCD = ABC
1 0 1 0 + 1 0 1 1 = 1 0 1 -(The dash indicates a missing variable)
ABCD + ABCD
0 1 0 1 + 0 1 1 0
We can combine the minterms above because theydiffer by a single bit.
The minterms below wont combine
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QuineQuine--McCluskey Method An ExampleMcCluskey Method An Example
1. Find all the prime implicants
= )14,10,9,8,7,6,5,2,1,0(),,,( mdcbaf
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 01016 01109 100110 1010
7 011114 1110
Group the minterms according to the numberof 1s in the minterm.
This way we only have tocompare minterms fromadjacent groups.
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QuineQuine--McCluskey Method An ExampleMcCluskey Method An Example
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 01016 01109 100110 1010
7 011114 1110
Column I Column II
Combininggroup 0 and
group 1:
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3434 / / 143143
QuineQuine--McCluskey Method An ExampleMcCluskey Method An Example
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 01016 01109 100110 1010
7 011114 1110
Column I Column II
0,1 000-
Combininggroup 0 and
group 1:
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3535 / / 143143
QuineQuine--McCluskey Method An ExampleMcCluskey Method An Example
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 01016 01109 100110 1010
7 011114 1110
Column I Column II
0,1 000-0,2 00-0
Combininggroup 0 and
group 1:
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3636 / / 143143
QuineQuine--McCluskey Method An ExampleMcCluskey Method An Example
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 01016 01109 100110 1010
7 011114 1110
Column I Column II
0,1 000-0,2 00-00,8 -000
Does it makesense tocombine group 0with group 2 or 3?
No, there are atleast two bits thatare different.
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3737 / / 143143
QuineQuine--McCluskey Method An ExampleMcCluskey Method An Example
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 01016 01109 100110 1010
7 011114 1110
Column I Column II
0,1 000-0,2 00-00,8 -000
Does it makesense to nocombine group 0with group 2 or 3?
No, there are atleast two bits thatare different.
Thus, next we combine group 1and group 2.
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3838 / / 143143
QuineQuine--McCluskey Method An ExampleMcCluskey Method An Example
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 01016 01109 100110 1010
7 011114 1110
Column I Column II
0,1 000-0,2 00-00,8 -0001,5 0-01
Combine group 1and group 2.
-
3939 / / 143143
QuineQuine--McCluskey Method An ExampleMcCluskey Method An Example
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 01016 01109 100110 1010
7 011114 1110
Column I Column II
0,1 000-0,2 00-00,8 -0001,5 0-01
Combine group 1and group 2.
-
4040 / / 143143
QuineQuine--McCluskey Method An ExampleMcCluskey Method An Example
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 01016 01109 100110 1010
7 011114 1110
Column I Column II
0,1 000-0,2 00-00,8 -0001,5 0-011,9 -001Combine group 1
and group 2.
-
4141 / / 143143
QuineQuine--McCluskey Method An ExampleMcCluskey Method An Example
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 01016 01109 100110 1010
7 011114 1110
Column I Column II
0,1 000-0,2 00-00,8 -0001,5 0-011,9 -001Combine group 1
and group 2.
-
4242 / / 143143
QuineQuine--McCluskey Method An ExampleMcCluskey Method An Example
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 01016 01109 100110 1010
7 011114 1110
Column I Column II
0,1 000-0,2 00-00,8 -0001,5 0-011,9 -001Combine group 1
and group 2.
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4343 / / 143143
QuineQuine--McCluskey Method An ExampleMcCluskey Method An Example
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 01016 01109 100110 1010
7 011114 1110
Column I Column II
0,1 000-0,2 00-00,8 -0001,5 0-011,9 -001Combine group 1
and group 2.
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4444 / / 143143
QuineQuine--McCluskey Method An ExampleMcCluskey Method An Example
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 01016 01109 100110 1010
7 011114 1110
Column I Column II
0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10
Combine group 1and group 2.
-
4545 / / 143143
QuineQuine--McCluskey Method An ExampleMcCluskey Method An Example
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 01016 01109 100110 1010
7 011114 1110
Column I Column II
0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10
Combine group 1and group 2.
-
4646 / / 143143
QuineQuine--McCluskey Method An ExampleMcCluskey Method An Example
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 01016 01109 100110 1010
7 011114 1110
Column I Column II
0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10
2,10 -010
Combine group 1and group 2.
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4747 / / 143143
QuineQuine--McCluskey Method An ExampleMcCluskey Method An Example
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 01016 01109 100110 1010
7 011114 1110
Column I Column II
0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10
2,10 -010
Combine group 1and group 2.
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4848 / / 143143
QuineQuine--McCluskey Method An ExampleMcCluskey Method An Example
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 01016 01109 100110 1010
7 011114 1110
Column I Column II
0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10
2,10 -010
Combine group 1and group 2.
-
4949 / / 143143
QuineQuine--McCluskey Method An ExampleMcCluskey Method An Example
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 01016 01109 100110 1010
7 011114 1110
Column I Column II
0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10
2,10 -0108,9 100-
Combine group 1and group 2.
-
5050 / / 143143
QuineQuine--McCluskey Method An ExampleMcCluskey Method An Example
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 01016 01109 100110 1010
7 011114 1110
Column I Column II
0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10
2,10 -0108,9 100-
8,10 10-0
Again, there isno need to tryto combine group1 with group 3.
Lets try to combinegroup 2 with group 3.
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QuineQuine--McCluskey Method An ExampleMcCluskey Method An Example
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 01016 01109 100110 1010
7 011114 1110
Column I Column II
0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10
2,10 -0108,9 100-
8,10 10-05,7 01-1
Combine group 2and group 3.
-
5252 / / 143143
QuineQuine--McCluskey Method An ExampleMcCluskey Method An Example
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 01016 01109 100110 1010
7 011114 1110
Column I Column II
0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10
2,10 -0108,9 100-
8,10 10-05,7 01-1
Combine group 2and group 3.
-
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QuineQuine--McCluskey Method An ExampleMcCluskey Method An Example
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 01016 01109 100110 1010
7 011114 1110
Column I Column II
0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10
2,10 -0108,9 100-
8,10 10-05,7 01-16,7 011-
Combine group 2and group 3.
-
5454 / / 143143
QuineQuine--McCluskey Method An ExampleMcCluskey Method An Example
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 01016 01109 100110 1010
7 011114 1110
Column I Column II
0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10
2,10 -0108,9 100-
8,10 10-05,7 01-16,7 011-
6,14 -110
Combine
group 2and group 3.
-
5555 / / 143143
QuineQuine--McCluskey Method An ExampleMcCluskey Method An Example
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 01016 01109 100110 1010
7 011114 1110
Column I Column II
0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10
2,10 -0108,9 100-
8,10 10-05,7 01-16,7 011-
6,14 -110
Combine group 2and group 3.
-
5656 / / 143143
QuineQuine--McCluskey Method An ExampleMcCluskey Method An Example
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 01016 01109 100110 1010
7 011114 1110
Column I Column II
0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10
2,10 -0108,9 100-
8,10 10-05,7 01-16,7 011-
6,14 -110
Combine group 2and group 3.
-
5757 / / 143143
QuineQuine--McCluskey Method An ExampleMcCluskey Method An Example
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 01016 01109 100110 1010
7 011114 1110
Column I Column II
0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10
2,10 -0108,9 100-
8,10 10-05,7 01-16,7 011-
6,14 -110
Combine group 2and group 3.
-
5858 / / 143143
QuineQuine--McCluskey Method An ExampleMcCluskey Method An Example
Column I Column II
0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10
2,10 -0108,9 100-
8,10 10-05,7 01-16,7 011-
6,14 -11010,14 1-10
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 01016 01109 100110 1010
7 011114 1110
We have nowcompleted thefirst step. Allminterms in column I wereincluded.
We can dividecolumn II intogroups.
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QuineQuine--McCluskey Method An ExampleMcCluskey Method An Example
Column I Column II
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 01016 01109 100110 1010
7 011114 1110
0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10
2,10 -0108,9 100-
8,10 10-05,7 01-16,7 011-
6,14 -11010,14 1-10
-
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QuineQuine--McCluskey Method An ExampleMcCluskey Method An Example
Column I Column II
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 01016 01109 100110 1010
7 011114 1110
0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10
2,10 -0108,9 100-
8,10 10-05,7 01-16,7 011-
6,14 -11010,14 1-10
Column III
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QuineQuine--McCluskey Method An ExampleMcCluskey Method An Example
Column I Column II
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 01016 01109 100110 1010
7 011114 1110
0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10
2,10 -0108,9 100-
8,10 10-05,7 01-16,7 011-
6,14 -11010,14 1-10
Column III
-
6262 / / 143143
QuineQuine--McCluskey Method An ExampleMcCluskey Method An Example
Column I Column II
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 01016 01109 100110 1010
7 011114 1110
0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10
2,10 -0108,9 100-
8,10 10-05,7 01-16,7 011-
6,14 -11010,14 1-10
Column III
-
6363 / / 143143
QuineQuine--McCluskey Method An ExampleMcCluskey Method An Example
Column I Column II
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 01016 01109 100110 1010
7 011114 1110
0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10
2,10 -0108,9 100-
8,10 10-05,7 01-16,7 011-
6,14 -11010,14 1-10
Column III
-
6464 / / 143143
QuineQuine--McCluskey Method An ExampleMcCluskey Method An Example
Column I Column II
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 01016 01109 100110 1010
7 011114 1110
0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10
2,10 -0108,9 100-
8,10 10-05,7 01-16,7 011-
6,14 -11010,14 1-10
Column III
-
6565 / / 143143
QuineQuine--McCluskey Method An ExampleMcCluskey Method An Example
Column I Column II
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 01016 01109 100110 1010
7 011114 1110
0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10
2,10 -0108,9 100-
8,10 10-05,7 01-16,7 011-
6,14 -11010,14 1-10
Column III
0,1,8,9 -00-
-
6666 / / 143143
QuineQuine--McCluskey Method An ExampleMcCluskey Method An Example
Column I Column II
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 01016 01109 100110 1010
7 011114 1110
0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10
2,10 -0108,9 100-
8,10 10-05,7 01-16,7 011-
6,14 -11010,14 1-10
Column III
0,1,8,9 -00-
-
6767 / / 143143
QuineQuine--McCluskey Method An ExampleMcCluskey Method An Example
Column I Column II
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 01016 01109 100110 1010
7 011114 1110
0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10
2,10 -0108,9 100-
8,10 10-05,7 01-16,7 011-
6,14 -11010,14 1-10
Column III
0,1,8,9 -00-
-
6868 / / 143143
QuineQuine--McCluskey Method An ExampleMcCluskey Method An Example
Column I Column II
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 01016 01109 100110 1010
7 011114 1110
0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10
2,10 -0108,9 100-
8,10 10-05,7 01-16,7 011-
6,14 -11010,14 1-10
Column III
0,1,8,9 -00-
-
6969 / / 143143
QuineQuine--McCluskey Method An ExampleMcCluskey Method An Example
Column I Column II
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 01016 01109 100110 1010
7 011114 1110
0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10
2,10 -0108,9 100-
8,10 10-05,7 01-16,7 011-
6,14 -11010,14 1-10
Column III
0,1,8,9 -00-
-
7070 / / 143143
QuineQuine--McCluskey Method An ExampleMcCluskey Method An Example
Column I Column II
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 01016 01109 100110 1010
7 011114 1110
0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10
2,10 -0108,9 100-
8,10 10-05,7 01-16,7 011-
6,14 -11010,14 1-10
Column III
0,1,8,9 -00-
-
7171 / / 143143
QuineQuine--McCluskey Method An ExampleMcCluskey Method An Example
Column I Column II
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 01016 01109 100110 1010
7 011114 1110
0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10
2,10 -0108,9 100-
8,10 10-05,7 01-16,7 011-
6,14 -11010,14 1-10
Column III
0,1,8,9 -00-
-
7272 / / 143143
QuineQuine--McCluskey Method An ExampleMcCluskey Method An Example
Column I Column II
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 01016 01109 100110 1010
7 011114 1110
0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10
2,10 -0108,9 100-
8,10 10-05,7 01-16,7 011-
6,14 -11010,14 1-10
Column III
0,1,8,9 -00-0,2,8,10 -0-0
-
7373 / / 143143
QuineQuine--McCluskey Method An ExampleMcCluskey Method An Example
Column I Column II
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 01016 01109 100110 1010
7 011114 1110
0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10
2,10 -0108,9 100-
8,10 10-05,7 01-16,7 011-
6,14 -11010,14 1-10
Column III
0,1,8,9 -00-0,2,8,10 -0-0
-
7474 / / 143143
QuineQuine--McCluskey Method An ExampleMcCluskey Method An Example
Column I Column II
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 01016 01109 100110 1010
7 011114 1110
0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10
2,10 -0108,9 100-
8,10 10-05,7 01-16,7 011-
6,14 -11010,14 1-10
Column III
0,1,8,9 -00-0,2,8,10 -0-0
0,8,1,9 -00-
-
7575 / / 143143
QuineQuine--McCluskey Method An ExampleMcCluskey Method An Example
Column I Column II
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 01016 01109 100110 1010
7 011114 1110
0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10
2,10 -0108,9 100-
8,10 10-05,7 01-16,7 011-
6,14 -11010,14 1-10
Column III
0,1,8,9 -00-0,2,8,10 -0-0
0,8,1,9 -00-
-
7676 / / 143143
QuineQuine--McCluskey Method An ExampleMcCluskey Method An Example
Column I Column II
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 01016 01109 100110 1010
7 011114 1110
0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10
2,10 -0108,9 100-
8,10 10-05,7 01-16,7 011-
6,14 -11010,14 1-10
Column III
0,1,8,9 -00-0,2,8,10 -0-0
0,8,1,9 -00-0,8,2,10 -0-0
-
7777 / / 143143
QuineQuine--McCluskey Method An ExampleMcCluskey Method An Example
Column I Column II
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 01016 01109 100110 1010
7 011114 1110
0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10
2,10 -0108,9 100-
8,10 10-05,7 01-16,7 011-
6,14 -11010,14 1-10
Column III
0,1,8,9 -00-0,2,8,10 -0-0
0,8,1,9 -00-0,8,2,10 -0-0
-
7878 / / 143143
QuineQuine--McCluskey Method An ExampleMcCluskey Method An Example
Column I Column II
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 01016 01109 100110 1010
7 011114 1110
0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10
2,10 -0108,9 100-
8,10 10-05,7 01-16,7 011-
6,14 -11010,14 1-10
Column III
0,1,8,9 -00-0,2,8,10 -0-0
0,8,1,9 -00-0,8,2,10 -0-0
-
7979 / / 143143
QuineQuine--McCluskey Method An ExampleMcCluskey Method An Example
Column I Column II
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 01016 01109 100110 1010
7 011114 1110
0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10
2,10 -0108,9 100-
8,10 10-05,7 01-16,7 011-
6,14 -11010,14 1-10
Column III
0,1,8,9 -00-0,2,8,10 -0-0
0,8,1,9 -00-0,8,2,10 -0-0
-
8080 / / 143143
QuineQuine--McCluskey Method An ExampleMcCluskey Method An Example
Column I Column II
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 01016 01109 100110 1010
7 011114 1110
0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10
2,10 -0108,9 100-
8,10 10-05,7 01-16,7 011-
6,14 -11010,14 1-10
Column III
0,1,8,9 -00-0,2,8,10 -0-0
0,8,1,9 -00-0,8,2,10 -0-0
-
8181 / / 143143
QuineQuine--McCluskey Method An ExampleMcCluskey Method An Example
Column I Column II
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 01016 01109 100110 1010
7 011114 1110
0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10
2,10 -0108,9 100-
8,10 10-05,7 01-16,7 011-
6,14 -11010,14 1-10
Column III
0,1,8,9 -00-0,2,8,10 -0-0
0,8,1,9 -00-0,8,2,10 -0-0
2,6,10,14 --10
-
8282 / / 143143
QuineQuine--McCluskey Method An ExampleMcCluskey Method An Example
Column I Column II
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 01016 01109 100110 1010
7 011114 1110
0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10
2,10 -0108,9 100-
8,10 10-05,7 01-16,7 011-
6,14 -11010,14 1-10
Column III
0,1,8,9 -00-0,2,8,10 -0-0
0,8,1,9 -00-0,8,2,10 -0-0
2,6,10,14 --102,10,6,14 --10
-
8383 / / 143143
QuineQuine--McCluskey Method An ExampleMcCluskey Method An Example
Column I Column II
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 01016 01109 100110 1010
7 011114 1110
0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10
2,10 -0108,9 100-
8,10 10-05,7 01-16,7 011-
6,14 -11010,14 1-10
Column III
0,1,8,9 -00-0,2,8,10 -0-0
0,8,1,9 -00-0,8,2,10 -0-0
2,6,10,14 --102,10,6,14 --10
-
8484 / / 143143
QuineQuine--McCluskey Method An ExampleMcCluskey Method An Example
Column I Column II
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 01016 01109 100110 1010
7 011114 1110
0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10
2,10 -0108,9 100-
8,10 10-05,7 01-16,7 011-
6,14 -11010,14 1-10
Column III
0,1,8,9 -00-0,2,8,10 -0-0
0,8,1,9 -00-0,8,2,10 -0-0
2,6,10,14 --102,10,6,14 --10
-
8585 / / 143143
QuineQuine--McCluskey Method An ExampleMcCluskey Method An Example
Column I Column II
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 01016 01109 100110 1010
7 011114 1110
0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10
2,10 -0108,9 100-
8,10 10-05,7 01-16,7 011-
6,14 -11010,14 1-10
Column III
0,1,8,9 -00-0,2,8,10 -0-0
0,8,1,9 -00-0,8,2,10 -0-0
2,6,10,14 --102,10,6,14 --10
No more combinationsare possible, thus westop here.
-
8686 / / 143143
QuineQuine--McCluskey Method An ExampleMcCluskey Method An Example
Column I Column II
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 01016 01109 100110 1010
7 011114 1110
0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10
2,10 -0108,9 100-
8,10 10-05,7 01-16,7 011-
6,14 -11010,14 1-10
Column III
0,1,8,9 -00-0,2,8,10 -0-0
0,8,1,9 -00-0,8,2,10 -0-0
2,6,10,14 --102,10,6,14 --10
We can eliminate repeatedcombinations
-
8787 / / 143143
QuineQuine--McCluskey Method An ExampleMcCluskey Method An Example
Column I Column II
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 01016 01109 100110 1010
7 011114 1110
0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10
2,10 -0108,9 100-
8,10 10-05,7 01-16,7 011-
6,14 -11010,14 1-10
Column III
0,1,8,9 -00-0,2,8,10 -0-0
2,6,10,14 --10
f = acd
Now we form f with theterms not checked
-
8888 / / 143143
QuineQuine--McCluskey Method An ExampleMcCluskey Method An Example
Column I Column II
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 01016 01109 100110 1010
7 011114 1110
0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10
2,10 -0108,9 100-
8,10 10-05,7 01-16,7 011-
6,14 -11010,14 1-10
Column III
0,1,8,9 -00-0,2,8,10 -0-0
2,6,10,14 --10
f = acd + abd
-
8989 / / 143143
QuineQuine--McCluskey Method An ExampleMcCluskey Method An Example
Column I Column II
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 01016 01109 100110 1010
7 011114 1110
0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10
2,10 -0108,9 100-
8,10 10-05,7 01-16,7 011-
6,14 -11010,14 1-10
Column III
0,1,8,9 -00-0,2,8,10 -0-0
2,6,10,14 --10
f = acd + abd + abc
-
9090 / / 143143
QuineQuine--McCluskey Method An ExampleMcCluskey Method An Example
Column I Column II
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 01016 01109 100110 1010
7 011114 1110
0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10
2,10 -0108,9 100-
8,10 10-05,7 01-16,7 011-
6,14 -11010,14 1-10
Column III
0,1,8,9 -00-0,2,8,10 -0-0
2,6,10,14 --10
f = acd + abd + abc + bc
-
9191 / / 143143
QuineQuine--McCluskey Method An ExampleMcCluskey Method An Example
Column I Column II
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 01016 01109 100110 1010
7 011114 1110
0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10
2,10 -0108,9 100-
8,10 10-05,7 01-16,7 011-
6,14 -11010,14 1-10
Column III
0,1,8,9 -00-0,2,8,10 -0-0
2,6,10,14 --10
f = acd + abd + abc + bc+ bd
-
9292 / / 143143
QuineQuine--McCluskey Method An ExampleMcCluskey Method An Example
Column I Column II
group 0
group 1
group 2
group 3
0 0000
1 00012 00108 1000
5 01016 01109 100110 1010
7 011114 1110
0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10
2,10 -0108,9 100-
8,10 10-05,7 01-16,7 011-
6,14 -11010,14 1-10
Column III
0,1,8,9 -00-0,2,8,10 -0-0
2,6,10,14 --10
f = acd + abd + abc + bc+ bd
+ cd
-
9393 / / 143143
QuineQuine--McCluskey Method An ExampleMcCluskey Method An Example
f = acd + abd + abc + bc
+ bd
+ cd
But, the form below is not minimized, using a Karnaugh map we can obtain:
a
1
b
c
d1
-
9494 / / 143143
QuineQuine--McCluskey Method An ExampleMcCluskey Method An Example
f = acd + abd + abc + bc
+ bd
+ cd
But, the form below is not minimized, using a Karnaugh map we can obtain:
a
1
1
b
c
d1
-
9595 / / 143143
QuineQuine--McCluskey Method An ExampleMcCluskey Method An Example
f = acd + abd + abc + bc
+ bd
+ cd
But, the form below is not minimized, using a Karnaugh map we can obtain:
a
1
1
1
b
c
d1
-
9696 / / 143143
QuineQuine--McCluskey Method An ExampleMcCluskey Method An Example
f = acd + abd + abc + bc
+ bd
+ cd
But, the form below is not minimized, using a Karnaugh map we can obtain:
a
1
1
1
1
1
1
b
c
d1
-
9797 / / 143143
QuineQuine--McCluskey Method An ExampleMcCluskey Method An Example
f = acd + abd + abc + bc
+ bd
+ cd
But, the form below is not minimized, using a Karnaugh map we can obtain:
a
1
1
1
1
1
1
1 1
b
c
d1
-
9898 / / 143143
QuineQuine--McCluskey Method An ExampleMcCluskey Method An Example
f = acd + abd + abc + bc
+ bd
+ cd
But, the form below is not minimized. Using a Karnaugh map we can obtain:
a
1
1
1
1
1
1
1 1 1
b
c
d1
-
9999 / / 143143
f = acd + abd + abc + bc
+ bd
+ cd
But, the form below is not minimized, using a Karnaugh map we can obtain:
a
1
1
1
1
1
1
1 1 1
b
c
d1
F = abd
QuineQuine--McCluskey Method An ExampleMcCluskey Method An Example
-
100100 / / 143143
f = acd + abd + abc + bc
+ bd
+ cd
But, the form below is not minimized, using a Karnaugh map we can obtain:
a
1
1
1
1
1
1
1 1 1
b
c
d1
F = abd + cd
QuineQuine--McCluskey Method An ExampleMcCluskey Method An Example
-
101101 / / 143143
f = acd + abd + abc + bc
+ bd
+ cd
But, the form below is not minimized, using a Karnaugh map we can obtain:
a
1
1
1
1
1
1
1 1 1
b
c
d1
F = abd + cd
+ bc
QuineQuine--McCluskey Method An ExampleMcCluskey Method An Example
-
102102 / / 143143
QuineQuine--McCluskey Method An ExampleMcCluskey Method An Example
f = acd + abd + abc + bc
+ bd
+ cd
What are the extra terms in the solution obtainedwith the Quine-McCluskey method?
a
1
1
1
1
1
1
1 1 1
b
c
d1
F = abd + cd
+ bc
Thus, we need a method to eliminate this redundant termsfrom the Quine-McCluskey solution.
-
103103 / / 143143
The Prime Implicant ChartThe Prime Implicant Chart
The prime implicant chart is the second part ofthe Quine-McCluskey procedure.
It is used to select a minimum set of prime implicants.
Similar to the Karnaugh map, we first selectthe essential prime implicants, and then weselect enough prime implicants to cover allthe minterms of the function.
-
104104 / / 143143
Prime Implicant Chart (Example)Prime Implicant Chart (Example)
0 1 2 5 6 7 8 9 10 14(0,1,8,9) bc X X X X (0,2,8,10) bd X X X X (2,6,10,14) cd X X X X (1,5) acd X X (5,7) abd X X (6,7) abc X X
Question: Given the prime implicant chart above,how can we identify the essential primeimplicants of the function?
mintermsP
rime
Impl
ican
ts
-
105105 / / 143143
Prime Implicant Chart (Example)Prime Implicant Chart (Example)
0 1 2 5 6 7 8 9 10 14(0,1,8,9) bc X X X X (0,2,8,10) bd X X X X (2,6,10,14) cd X X X X (1,5) acd X X (5,7) abd X X (6,7) abc X X
Similar to the Karnaugh map, all we have to do is to look for minterms that are covered by a singleterm.
mintermsP
rime
Impl
ican
ts
-
106106 / / 143143
Prime Implicant Chart (Example)Prime Implicant Chart (Example)
0 1 2 5 6 7 8 9 10 14(0,1,8,9) bc X X X X (0,2,8,10) bd X X X X (2,6,10,14) cd X X X X (1,5) acd X X (5,7) abd X X (6,7) abc X X
Once a term is included in the solution, all theminterms covered by that term are covered.
Therefore we may now mark the covered mintermsand find terms that are no longer useful.
mintermsP
rime
Impl
ican
ts
-
107107 / / 143143
Prime Implicant Chart (Example)Prime Implicant Chart (Example)
0 1 2 5 6 7 8 9 10 14(0,1,8,9) bc X X X X (0,2,8,10) bd X X X X (2,6,10,14) cd X X X X (1,5) acd X X (5,7) abd X X (6,7) abc X X
mintermsP
rime
Impl
ican
ts
-
108108 / / 143143
Prime Implicant Chart (Example)Prime Implicant Chart (Example)
0 1 2 5 6 7 8 9 10 14(0,1,8,9) bc X X X X (0,2,8,10) bd X X X X (2,6,10,14) cd X X X X (1,5) acd X X (5,7) abd X X (6,7) abc X X
As we have not covered all the minterms withessential prime implicants, we must chooseenough non-essential prime implicants to cover the remaining minterms.
mintermsP
rime
Impl
ican
ts
-
109109 / / 143143
Prime Implicant Chart (Example)Prime Implicant Chart (Example)
0 1 2 5 6 7 8 9 10 14(0,1,8,9) bc X X X X (0,2,8,10) bd X X X X (2,6,10,14) cd X X X X (1,5) acd X X (5,7) abd X X (6,7) abc X X
What strategy should we use to find a minimumcover for the remaining minterms?
mintermsP
rime
Impl
ican
ts
-
110110 / / 143143
Prime Implicant Chart (Example)Prime Implicant Chart (Example)
0 1 2 5 6 7 8 9 10 14(0,1,8,9) bc X X X X (0,2,8,10) bd X X X X (2,6,10,14) cd X X X X (1,5) acd X X (5,7) abd X X (6,7) abc X X
We choose first prime implicants that cover themost minterms. Should this strategy always work??
mintermsP
rime
Impl
ican
ts
-
111111 / / 143143
Prime Implicant Chart (Example)Prime Implicant Chart (Example)
0 1 2 5 6 7 8 9 10 14(0,1,8,9) bc X X X X (0,2,8,10) bd X X X X (2,6,10,14) cd X X X X (1,5) acd X X (5,7) abd X X (6,7) abc X X
Therefore our minimum solution is:
f(a,b,c,d) = bc
+ cd
+ abd
mintermsP
rime
Impl
ican
ts
-
112112 / / 143143
Cyclic Prime Implicant ChartCyclic Prime Implicant Chart
F(a,b,c) =
m(0, 1, 2, 5, 6, 7)
0 000 1 001 2 010 5 101 6 110 7 111
0,1 00-0,2 0-01,5 -012,6 -105,7 1-16,7 11-
0 1 2 5 6 7(0,1) ab X X (0,2) ac X X (1,5) bc X X (2,6) bc X X (5,7) ac X X(6,7) ab X X
Which ones are the essential prime implicants in this chart?
There is no essential prime implicants, how we proceed?
minterms
Prim
e Im
plic
ants
-
113113 / / 143143
Cyclic Prime Implicant ChartCyclic Prime Implicant Chart
F(a,b,c) =
m(0, 1, 2, 5, 6, 7)
0 000 1 001 2 010 5 101 6 110 7 111
0,1 00-0,2 0-01,5 -012,6 -105,7 1-16,7 11-
0 1 2 5 6 7(0,1) ab X X (0,2) ac X X (1,5) bc X X (2,6) bc X X (5,7) ac X X(6,7) ab X X
Also, all implicants cover the same number of minterms. We will have to proceed by trial and error.
minterms
Prim
e Im
plic
ants
F(a,b,c) = ab
-
114114 / / 143143
Cyclic Prime Implicant ChartCyclic Prime Implicant Chart
F(a,b,c) =
m(0, 1, 2, 5, 6, 7)
0 000 1 001 2 010 5 101 6 110 7 111
0,1 00-0,2 0-01,5 -012,6 -105,7 1-16,7 11-
0 1 2 5 6 7(0,1) ab X X (0,2) ac X X (1,5) bc X X (2,6) bc X X (5,7) ac X X(6,7) ab X X
Also, all implicants cover the same number of minterms. We will have to proceed by trial and error.
minterms
Prim
e Im
plic
ants
F(a,b,c) = ab
+ bc
-
115115 / / 143143
Cyclic Prime Implicant ChartCyclic Prime Implicant Chart
F(a,b,c) =
m(0, 1, 2, 5, 6, 7)
0 000 1 001 2 010 5 101 6 110 7 111
0,1 00-0,2 0-01,5 -012,6 -105,7 1-16,7 11-
0 1 2 5 6 7(0,1) ab X X (0,2) ac X X (1,5) bc X X (2,6) bc X X (5,7) ac X X(6,7) ab X X
Thus, we get the minimization:
F(a,b,c) = ab
+ bc
+ ac
minterms
Prim
e Im
plic
ants
-
116116 / / 143143
Cyclic Prime Implicant ChartCyclic Prime Implicant Chart
F(a,b,c) =
m(0, 1, 2, 5, 6, 7)
0 000 1 001 2 010 5 101 6 110 7 111
0,1 00-0,2 0-01,5 -012,6 -105,7 1-16,7 11-
0 1 2 5 6 7(0,1) ab X X (0,2) ac X X (1,5) bc X X (2,6) bc X X (5,7) ac X X(6,7) ab X X
Lets try another set of prime implicants.
minterms
Prim
e Im
plic
ants
-
117117 / / 143143
Cyclic Prime Implicant ChartCyclic Prime Implicant Chart
F(a,b,c) =
m(0, 1, 2, 5, 6, 7)
0 000 1 001 2 010 5 101 6 110 7 111
0,1 00-0,2 0-01,5 -012,6 -105,7 1-16,7 11-
0 1 2 5 6 7(0,1) ab X X (0,2) ac X X (1,5) bc X X (2,6) bc X X (5,7) ac X X(6,7) ab X X
Lets try another set of prime implicants.
minterms
Prim
e Im
plic
ants
F(a,b,c) = ac
-
118118 / / 143143
Cyclic Prime Implicant ChartCyclic Prime Implicant Chart
F(a,b,c) =
m(0, 1, 2, 5, 6, 7)
0 000 1 001 2 010 5 101 6 110 7 111
0,1 00-0,2 0-01,5 -012,6 -105,7 1-16,7 11-
0 1 2 5 6 7(0,1) ab X X (0,2) ac X X (1,5) bc X X (2,6) bc X X (5,7) ac X X(6,7) ab X X
Lets try another set of prime implicants.
minterms
Prim
e Im
plic
ants
F(a,b,c) = ac + bc
-
119119 / / 143143
Cyclic Prime Implicant ChartCyclic Prime Implicant Chart
F(a,b,c) =
m(0, 1, 2, 5, 6, 7)
0 000 1 001 2 010 5 101 6 110 7 111
0,1 00-0,2 0-01,5 -012,6 -105,7 1-16,7 11-
0 1 2 5 6 7(0,1) ab X X (0,2) ac X X (1,5) bc X X (2,6) bc X X (5,7) ac X X(6,7) ab X X
Lets try another set of prime implicants.
minterms
Prim
e Im
plic
ants
F(a,b,c) = ac + bc+ ab
-
120120 / / 143143
Cyclic Prime Implicant ChartCyclic Prime Implicant Chart
F(a,b,c) =
m(0, 1, 2, 5, 6, 7)
0 000 1 001 2 010 5 101 6 110 7 111
0,1 00-0,2 0-01,5 -012,6 -105,7 1-16,7 11-
0 1 2 5 6 7(0,1) ab X X (0,2) ac X X (1,5) bc X X (2,6) bc X X (5,7) ac X X(6,7) ab X X
This time we obtain:
F(a,b,c) = ac + bc+ ab
minterms
Prim
e Im
plic
ants
-
121121 / / 143143
Cyclic Prime Implicant ChartCyclic Prime Implicant Chart
Which minimal form is better?
F(a,b,c) = ab
+ bc
+ ac
F(a,b,c) = ac + bc+ ab
Depends on what terms we must form for otherfunctions that we must also implement.
Often we are interested in examining all minimalforms for a given function.
Thus we need an algorithm to do so.
-
122122 / / 143143
PetrickPetricks Methods Method
S. R. Petrick. A direct determination of the irredundantforms of a boolean function from the set of prime implicants. Technical Report AFCRC-TR-56-110, Air Force Cambridge Research Center, Cambridge, MA, April, 1956.
Goal: Given a prime implicant chart, determineall minimum sum-of-products solutions.
-
123123 / / 143143
PetrickPetricks Methods Method An ExampleAn Example
0 1 2 5 6 7P1 (0,1) ab X X P2 (0,2) ac X X P3 (1,5) bc X X P4 (2,6) bc X X P5 (5,7) ac X XP6 (6,7) ab X X
Step 1: Label all the rows in the chart.
Step 2: Form a logic function P with the logic variables P1
, P2
, P3
that is true whenall the minterms in the chart are covered.
minterms
Prim
e Im
plic
ants
-
124124 / / 143143
PetrickPetricks Methods Method An ExampleAn Example
0 1 2 5 6 7P1 (0,1) ab X X P2 (0,2) ac X X P3 (1,5) bc X X P4 (2,6) bc X X P5 (5,7) ac X XP6 (6,7) ab X X
The first column has an X in rows P1
and P2
. Therefore we must include one of these rowsin order to cover minterm 0. Thus the followingterm must be in P:
(P1
+ P2
)
minterms
Prim
e Im
plic
ants
-
125125 / / 143143
PetrickPetricks Methods Method An ExampleAn Example
0 1 2 5 6 7P1 (0,1) ab X X P2 (0,2) ac X X P3 (1,5) bc X X P4 (2,6) bc X X P5 (5,7) ac X XP6 (6,7) ab X X
Following this technique, we obtain:
P = (P1
+ P2
) (P1
+ P3
) (P2
+ P4
) (P3
+ P5
) (P4
+ P6
) (P5
+ P6
)
P = (P1
+ P2
) (P1
+ P3
) (P4
+ P2
) (P5
+ P3
) (P4
+ P6
) (P5
+ P6
)
P = (P1
+ P2
) (P1
+ P3
) (P4
+ P2
) (P4
+ P6
) (P5
+ P3
) (P5
+ P6
)
P = (P1
+ P2 P3
) (P4
+ P2 P6
) (P5
+ P3 P6
)
minterms
Prim
e Im
plic
ants
-
126126 / / 143143
PetrickPetricks Methods Method An ExampleAn Example
P = (P1
+ P2
) (P1
+ P3
) (P2
+ P4
) (P3
+ P5
) (P4
+ P6
) (P5
+ P6
)
P = (P1
+ P2
) (P1
+ P3
) (P4
+ P2
) (P5
+ P3
) (P4
+ P6
) (P5
+ P6
)
P = (P1
+ P2
) (P1
+ P3
) (P4
+ P2
) (P4
+ P6
) (P5
+ P3
) (P5
+ P6
)
P = (P1
+ P2 P3
) (P4
+ P2 P6
) (P5
+ P3 P6
)
P = (P1 P4
+ P1
P2 P6
+ P2 P3 P4
+ P2 P3 P6
) (P5
+ P3 P6
)
P = P1 P4 P5
+ P1
P2 P5 P6
+ P2 P3 P4 P5
+ P2 P3 P5 P6+ P1 P3 P4 P6
+ P1
P2 P3 P6
+ P2 P3 P4 P6
+ P2 P3 P6
P = P1 P4 P5
+ P1
P2 P5 P6
+ P2 P3 P4 P5
+ P1 P4 P3 P6
+ P2 P3 P6
-
127127 / / 143143
PetrickPetricks Methods Method
An ExampleAn Example
P = P1 P4 P5
+ P1
P2 P5 P6
+ P2 P3 P4 P5
+ P1 P4 P3 P6
+ P2 P3 P6
This expression says that to cover all the mintermswe must include the terms in line P1
and line P4
and line P5
, or we must include line P1
, and line P2
, and line P5
, and line P6
, or
Considering that all the terms P1
, P2
,
have the samecost, how many minimal forms the function has?
The two minimal forms are P1 P4 P5 and P2 P3 P6:
F = ab
+ bc
+ ac F = ac
+ bc + ab
-
128128 / / 143143
Universal GatesUniversal Gates
One Type
Use as many as you need (quantity), but one type only.
Perform Basic Operations
AND, OR, and NOT
NAND Gate
NOT-AND functions
OR function can be obtained from AND by Demorgans
NOR Gate
NOT-OR functions (AND by Demorgans)
-
129129 / / 143143
Universal GatesUniversal Gates
NAND Gate
NOT:
AND:
OR: DeMorgans
-
130130 / / 143143
Universal GatesUniversal Gates
NOR Gate
NOT:
OR:
AND: DeMorgans
-
131131 / / 143143
NAND & NOR ImplementationNAND & NOR Implementation
Two-Level Implementation
-
132132 / / 143143
NAND & NOR ImplementationNAND & NOR Implementation
Two-Level Implementation
-
133133 / / 143143
NAND & NOR ImplementationNAND & NOR Implementation
Multilevel NAND Implementation
CDBABC
F
CDBABC
F
-
134134 / / 143143
NAND & NOR ImplementationNAND & NOR Implementation
Multilevel NOR Implementation
-
135135 / / 143143
Gate ShapesGate Shapes
AND
OR
NAND
NOR
-
136136 / / 143143
Other ImplementationsOther Implementations
AND-OR-Invert
OR-AND-Invert
-
137137 / / 143143
Implementations SummaryImplementations Summary
Sum Of Products:
AND-OR
AND-OR-Invert ==
AND-NOR ==
NAND-AND
Products Of Sums
OR-AND
OR-AND-Invert ==
OR-NAND ==
NOR--OR
-
138138 / / 143143
ExclusiveExclusive--OROR
XORF = x
y = x y + x y
XNORF = x
y =
x y = x y + x y
-
139139 / / 143143
ExclusiveExclusive--OROR
Identities
x
0 =
x
x
1 =
x
x
x =
0
x
x =
1
x
y =
x
y =
x
y
Commutative & Associative
x
y =
y
x
( x
y )
z =
x
( y
z ) =
x
y
z
x y XOR0 0 00 1 11 0 11 1 0
-
140140 / / 143143
ExclusiveExclusive--OR FunctionsOR Functions
Odd FunctionF = x
y
z
F = (1, 2, 4, 7)
Even FunctionF = x
y
z
F = (0, 3, 5, 6)
x y z XOR XNOR0 0 0 0 10 0 1 1 00 1 0 1 00 1 1 0 11 0 0 1 01 0 1 0 11 1 0 0 11 1 1 1 0
y zx 00 01 11 10
0 0 1 0 1
1 1 0 1 0
-
141141 / / 143143
ParityParity
1010
1010
1000
10101
10101
10001
Parity Generator
Parity Checker
-
142142 / / 143143
Parity GeneratorParity Generator
Odd Parity
Even Parity
1010
1
Odd number of 1s
1010
0
Even number of 1s
1010
1010
-
143143 / / 143143
Parity CheckerParity Checker
Odd Parity
Even Parity
ErrorCheck
1010
1
1010
0Error
Check
CE211 Digital SystemsKarnaugh MapTwo-variable MapTwo-variable MapTwo-variable MapThree-variable MapThree-variable MapThree-variable MapThree-variable MapThree-variable MapFour-variable MapFour-variable MapFour-variable MapFour-variable MapFour-variable MapFour-variable MapFour-variable MapFour-variable MapFive-variable MapFive-variable MapImplicantsPrime ImplicantsEssential Prime ImplicantsProduct of Sums SimplificationDont-Care ConditionDont-Care ConditionDont-Care ConditionDont-Care ConditionQuineMcCluskey Tabular Method MotivationQuine-McCluskeyOutline of the Quine-McCluskey MethodDetermination of Prime ImplicantsQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleSlide Number 100Slide Number 101Slide Number 102Quine-McCluskey Method An ExampleThe Prime Implicant ChartPrime Implicant Chart (Example)Prime Implicant Chart (Example)Prime Implicant Chart (Example)Prime Implicant Chart (Example)Prime Implicant Chart (Example)Prime Implicant Chart (Example)Prime Implicant Chart (Example)Prime Implicant Chart (Example)Cyclic Prime Implicant ChartCyclic Prime Implicant ChartCyclic Prime Implicant ChartCyclic Prime Implicant ChartCyclic Prime Implicant ChartCyclic Prime Implicant ChartCyclic Prime Implicant ChartCyclic Prime Implicant ChartCyclic Prime Implicant ChartCyclic Prime Implicant ChartPetricks MethodPetricks MethodAn ExamplePetricks MethodAn ExamplePetricks MethodAn ExamplePetricks MethodAn ExamplePetricks MethodAn ExampleUniversal GatesUniversal GatesUniversal GatesNAND & NOR ImplementationNAND & NOR ImplementationNAND & NOR ImplementationNAND & NOR ImplementationGate ShapesOther ImplementationsImplementations SummaryExclusive-ORExclusive-ORExclusive-OR FunctionsParityParity GeneratorParity Checker