K-Map (2B)€¦ · 18/09/2013 · K-Map (2B) 3 Young Won Lim 9/18/13 K-Map 4 variables (1) 0 0 1 1...
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Young Won Lim 9/18/13 K-Map (2B)
Transcript of K-Map (2B)€¦ · 18/09/2013 · K-Map (2B) 3 Young Won Lim 9/18/13 K-Map 4 variables (1) 0 0 1 1...
Young Won Lim9/18/13
K-Map (2B)
Young Won Lim9/18/13
Copyright (c) 2011-2013 Young W. Lim.
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K-Map (2B) 3 Young Won Lim9/18/13
K-Map 4 variables (1)
0 0 1 1
0 0 1 0
0 0 0 1
0 0 0 0
0 1 1 1
0 1 1 0
0 1 0 1
0 1 0 0
1 0 1 1
1 0 1 0
1 0 0 1
1 0 0 0
1 1 1 1
1 1 1 0
1 1 0 1
1 1 0 0
3
2
0
1
7
6
4
5
11
10
8
9
15
14
12
131000 1001 1011 1010
1100 1101 1111 1110
0100 0101 0111 0110
0000 0001 0011 0010
2310
6754
101198
14151312
x y z w
x y z w
x y z w
x y zw
x y z w
x y z w
x y z w
x y zw
x y z w
x y z w
x y z w
x y zw
x y z w
x y z w
x y z w
x y zw
z=1z=0
w=1w=0 w=0
x=1
x=0
y=1
y=0
y=0
1011010000
01
10
11
index minterms
K-Map (2B) 4 Young Won Lim9/18/13
K-Map 4 variables (2)
0 0 1 1
0 0 1 0
0 0 0 1
0 0 0 0
0 1 1 1
0 1 1 0
0 1 0 1
0 1 0 0
1 0 1 1
1 0 1 0
1 0 0 1
1 0 0 0
1 1 1 1
1 1 1 0
1 1 0 1
1 1 0 0
3
2
0
1
7
6
4
5
11
10
8
9
15
14
12
131000 1001 1011 1010
1100 1101 1111 1110
0100 0101 0111 0110
0000 0001 0011 0010
2310
6754
101198
14151312
x y z w
x y z w
x y z w
x y zw
x y z w
x y z w
x y z w
x y zw
x y z w
x y z w
x y z w
x y zw
x y z w
x y z w
x y z w
x y zw
z=1z=0
w=1w=0 w=0
x=1
x=0
y=1
y=0
y=0
index minterms
K-Map (2B) 5 Young Won Lim9/18/13
K-Map 3 variables (1)
2310
6754
y=1y=0z=1z=0 z=0
x=0
x=1
0 1 1
0 1 0
0 0 1
0 0 0
1 1 1
1 1 0
1 0 1
1 0 0
3
2
0
1
7
6
4
5
x y z
x y z
x y z
x y z
x y z
x y z
x y z
x y z
2310
6754
10110100
0
1
x y zindex minterms
K-Map (2B) 6 Young Won Lim9/18/13
K-Map 3 variables (2)
0 1 1
0 1 0
0 0 1
0 0 0
1 1 1
1 1 0
1 0 1
1 0 0
3
2
0
1
7
6
4
5
x y z
x y z
x y z
x y z
x y z
x y z
x y z
x y z
x y
x y
x y
x y
x y z + x y z = x y ( z+z) = x y
2310
6754
y=1y=0
z=1z=0 z=0
x=0
x=1
10110100
0
1
x y x y
x y x y
index minterms
x y z
21
a group of 2 minterms
K-Map (2B) 7 Young Won Lim9/18/13
K-Map 3 variables (3)
0 1 1
0 1 0
0 0 1
0 0 0
1 1 1
1 1 0
1 0 1
1 0 0
3
2
0
1
7
6
4
5
x y z
x y z
x y z
x y z
x y z
x y z
x y z
x y z
x z
x z
x z
x z
x y z + x y z = x z( y+y) = x z
2310
6754
y=1y=0
z=1z=0 z=0
x=0
x=1
10110100
0
1
x z x z
x z x z
index minterms
x y z
21
a group of 2 minterms
K-Map (2B) 8 Young Won Lim9/18/13
K-Map 3 variables (4)
0 1 1
0 1 0
0 0 1
0 0 0
1 1 1
1 1 0
1 0 1
1 0 0
3
2
0
1
7
6
4
5
x y z
x y z
x y z
x y z
x y z
x y z
x y z
x y z
y z
y z
y z
y z
x y z + x y z = y z ( x+x ) = y z
2310
6754
y=1y=0
z=1z=0 z=0
x=0
x=1
10110100
0
1
y z y z y z y z
index minterms
x y z
21
a group of 2 minterms
K-Map (2B) 9 Young Won Lim9/18/13
K-Map 3 variables (5)
0 1 1
0 1 0
0 0 1
0 0 0
1 1 1
1 1 0
1 0 1
1 0 0
3
2
0
1
7
6
4
5
x y z
x y z
x y z
x y z
x y z
x y z
x y z
x y z
x
x
2310
6754
y=1y=0
z=1z=0 z=0
x=0
x=1
10110100
0
1 x
x
index minterms
x y z
22
a group of 4 minterms
K-Map (2B) 10 Young Won Lim9/18/13
K-Map 3 variables (5)
0 1 1
0 1 0
0 0 1
0 0 0
1 1 1
1 1 0
1 0 1
1 0 0
3
2
0
1
7
6
4
5
x y z
x y z
x y z
x y z
x y z
x y z
x y z
x y z
y
y
2310
6754
y=1y=0
z=1z=0 z=0
x=0
x=1
10110100
0
1
y y
index minterms
x y z
22
a group of 4 minterms
K-Map (2B) 11 Young Won Lim9/18/13
K-Map 3 variables (5)
0 1 1
0 1 0
0 0 1
0 0 0
1 1 1
1 1 0
1 0 1
1 0 0
3
2
0
1
7
6
4
5
x y z
x y z
x y z
x y z
x y z
x y z
x y z
x y z
z
z
2310
6754
y=1y=0
z=1z=0 z=0
x=0
x=1
10110100
0
1 z z
index minterms
x y z
22
a group of 4 minterms
K-Map (2B) 12 Young Won Lim9/18/13
K-Map, minterms, and Maxterms
0 1 1
0 1 0
0 0 1
0 0 0
1 1 1
1 1 0
1 0 1
1 0 0
3
2
0
1
7
6
4
5
x y z
x y z
x y z
x y z
x y z
x y z
x y z
x y z
2310
6754
10110100
0
1
x y zindex minterms
m0 m1 m2m3
m4 m5m6m7
2310
6754
10110100
0
1
M 0 M 1 M 2M 3
M 4 M 5 M 6M 7
x y z
Each rectangle is associated with a minterm or a maxterm which represents a particular input variable conditions.
In this table, output function value is overlaid
K-Map (2B) 13 Young Won Lim9/18/13
1
0
1
0
0
0
0
1
Boolean Function with minterms
0 1 1
0 1 0
0 0 1
0 0 0
1 1 1
1 1 0
1 0 1
1 0 0
x y z
3
2
0
1
7
6
4
5
F
1
11
1F
2310
6754
x
y
z
1 1
1F = x z + x y z
F=1 when xz = 1
F=1 when xyz = 1
2310
6754
10110100
0
1
m0 m1 m2m3
m4 m5m6m7
an overlaid table
a simplified function
K-Map (2B) 14 Young Won Lim9/18/13
Boolean Function with Maxterms
1
0
1
0
0
0
0
1
0 1 1
0 1 0
0 0 1
0 0 0
1 1 1
1 1 0
1 0 1
1 0 0
x y z
3
2
0
1
7
6
4
5
F0
0
0
0
0
F
0
2310
6754
0 0
0 0 0F = ( x+ z)( y+z)(x+z)
x
y
z
F=0 when x+z=0
F=0 when y+z=0F=0 when x+z=0
2310
6754
M 0 M 1 M 2M 3
M 4 M 5 M 6M 7
an overlaid table
a simplified function
K-Map (2B) 15 Young Won Lim9/18/13
Implicant
P ⇒ Fimplies
F takes the value 1, whenever P equals 1
assuming P is a product term in a sum of products
f (x , y , z ,w) = xy + yz + w
xy f (x , y , z ,w) = xy + yz + w⇒
xyz f (x , y , z ,w) = xy + yz + w⇒
xyzw f (x , y , z ,w) = xy + yz + w⇒
(x=1, y=1)
(x=1, y=1, z=1)
(x=1, y=1, z=1, w=1)
general, reduced
K-Map (2B) 16 Young Won Lim9/18/13
Prime Implicant
Prime Implicant: An implicant that is minimal
The removal of any literal from P results in a non-implicant for F
xy f (x , y , z ,w) = xy + yz + w⇒
xyz f (x , y , z ,w) = xy + yz + w⇒
xyzw f (x , y , z ,w) = xy + yz + w⇒
(x=1, y=1)
(x=1, y=1, z=1)
(x=1, y=1, z=1, w=1)
general, reducedPrime Implicant
xyzw ⇒ f
xyzw ⇒ f
xyzw ⇒ f
xyzw ⇒ f
xyzw ⇒ f
xyzw ⇒ f
xyzw ⇒ f
xyzw ⇒ f
xyzw ⇒ f
xyzw ⇒ f
A process of expanding terms
general, reduced
K-Map (2B) 17 Young Won Lim9/18/13
z=1
y=1
x=1
Expanding Terms
1 1
1 1 1 1
1 1 1
1 1
w=1
xyzw ⇒ f
xyzw ⇒ f
xyzw ⇒ f
xyzw ⇒ f
xyzw ⇒ f
xyzw ⇒ f
xyzw ⇒ f
xyzw ⇒ f
xyzw ⇒ f
xyzw ⇒ f
z=1
y=1
x=1
1 1
1 1 1 1
1 1 1
1 1
w=1
z=1
y=1
x=1
1 1
1 1 1 1
1 1 1
1 1
w=1
Prime Implicant Prime Implicant Prime Implicant
K-Map (2B) 18 Young Won Lim9/18/13
z=1
y=1
x=1
Prime Implicant Example
1 1
1 1 1 1
1 1 1
1 1
w=1
z=1
y=1
x=1
1 1
1 1 1 1
1 1 1
1 1
w=1
z=1y=1
x=1
1 1
1 1 1 1
1 1 1
1 1
w=1
z=1
y=1
x=1
1 1
1 1 1 1
1 1 1
1 1
w=1
z=1
y=1
x=1
1 1
1 1 1 1
1 1 1
1 1
w=1
z=1
y=1
x=1
1 1
1 1 1 1
1 1 1
1 1
w=1
Prime Implicant Prime Implicant Prime Implicant
Prime Implicant Prime Implicant Prime Implicant
K-Map (2B) 19 Young Won Lim9/18/13
Essential Prime Implicant
Essential Prime Implicant: prime implicants
that cover an output of the function that no combination of other implicants is able to cover
z=1
y=1
x=1
1 1
1 1 1 1
1 1 1
1 1
w=1
z=1
y=1
x=1
1 1
1 1 1 1
1 1 1
1 1
w=1
z=1
y=1
x=1
1 1
1 1 1 1
1 1 1
1 1
w=1
Essential Prime Implicant Essential Prime Implicant Essential Prime Implicant
K-Map (2B) 20 Young Won Lim9/18/13
Implicant
P ⇒ Fimplies
F takes the value 1, whenever P equals 1
assuming P is a product term in a sum of products
f (x , y , z ,w) = xy + yz + w
xy f (x , y , z ,w) = xy + yz + w⇒
xyz f (x , y , z ,w) = xy + yz + w⇒
xyzw f (x , y , z ,w) = xy + yz + w⇒
implicants
General,Reduced
K-Map (2B) 21 Young Won Lim9/18/13
2's Complement
K-Map (2B) 22 Young Won Lim9/18/13
Decimal to Binary (1)
K-Map (2B) 23 Young Won Lim9/18/13
Decimal to Binary (2)
K-Map (2B) 24 Young Won Lim9/18/13
Laplace Equation
Decimal
K-Map (2B) 25 Young Won Lim9/18/13
Laplace Equation
K-Map (2B) 26 Young Won Lim9/18/13
Laplace Equation
Young Won Lim9/18/13
References
[1] http://en.wikipedia.org/[2] http://planetmath.org/[3] M.L. Boas, “Mathematical Methods in the Physical Sciences”
