Key Observation

1
Key Observation
Adjacencies in the K-Map
111
011
B
BC
00
01
11
10
A
010
110
0
000
010
100
001
B
001
101
1
101
C
000
C
100
A
Any two adjacant cells in the K-map differ in
exactly one variable. The uniting theorem can be
applied when adjacent cells contain (function
1) to eliminat the changing variable.
2
Design Examples
Two Bit Comparator
Truth Table
Block Diagram
Design Steps 1- Simplify to reduce cost (three
4-variable K-maps, one for each output) 2-
Implement using a suitable design style
(e.g. 2-level AND-OR, NAND-NAND, or multilevel
techniques)
3
Design Examples
Two Bit Comparator (continued)
F1 A' B' C' D' A' B C' D A B C D A
B' C D' F2 A' B' D A' C F3 B C' D'
A C' A B D'
f1 AC(BDbd) ac(bdBD) (ACac) (bdBD)
(a c) (b d)
4
Design Examples
Two-Bit Adder
Truth Table
Block Diagram
5
Design Example
Two-Bit Adder (Continued)
X A C B C D A B D Z B D' B' D
B xor D Y A' B' C A B' C' A' B C' D
A' B C D' A B C' D' A B C D
6
5-Variable K-Maps
Constructed from two 4-variable K-Maps.
A 0
A 1
f bCDE Bd
E
g ABe Bce bE
7
5-Variable K-Maps
Another View
(A,B,C,D,E) Sm(2,5,7,8,10,13,15,17,19,21,23,24,
29 31)
(A,B,C,D,E) C E A B' E B C' D' E'
A' C' D E'
8
6-Variable K-Maps
(A,B,C,D,E,F) Sm(2,8,10,18,24, 26,34,37,42,45,5
0, 53,58,61)
9
6-Variable K-Maps
(A,B,C,D,E,F) Sm(2,8,10,18,24, 26,34,37,42,45,5
0, 53,58,61)
D' E F' A D E' F A' C D' F'
10
Definitions
Implicant Single 1 entry or any group of 1s
that can be combined together in
a K-map to form a product term. (Dual)
Single 0 entry or any group of 0s that can be
combined together in a K-map to
form a sum term. Prime Implicant an implicant
that cannot be combined with another
implicant to eliminate a
term Essential Prime Implicant a prime
implicant that contains a 1 entry
not covered by any other implicant.
Example
C
6 Prime Implicants
00
01
11
10
AB
A' B C , C D, A D , B C' D , A C , AB D'
0
1
1
0
00
1
1
1
0
01
essential
B
1
0
1
1
11
A
Minimum cover A D A C B C D
0
0
1
1
10
D
11
Definitions
Implicant Single 1 entry or any group of 1s
that can be combined together in
a K-map to form a product term. (Dual)
Single 0 entry or any group of 0s that can be
combined together in a K-map to
form a sum term. Prime Implicant an implicant
that cannot be combined with another
implicant to eliminate a
term Essential Prime Implicant a prime
implicant that contains a 1 entry
not covered by any other implicant.
Example
C
00
01
11
10
AB
0
1
1
0
00
1
1
1
0
01
B
1
0
1
1
11
A
0
0
1
1
10
D
12
Illustrating the Definitions
Prime Implicants
B D, C D, A C, B' C
essential
Essential primes form the minimum cover
5 Prime Implicants
B D, A B C', A C D, A' B C, A' C' D
essential
Essential implicants form minimum cover
13
Illustrating the Definitions
C
Prime Implicants
00
01
11
10
AB
A D, AB, A C, BD
0
0
0
0
00
essential
0
1
1
0
01
B
1
1
1
1
11
Essential primes form the minimum cover
A
1
0
1
1
10
D
Try the other axis
5 Prime Implicants
B D, A B C', A C D, A' B C, A' C' D
essential
Essential implicants form minimum cover
14
Quine-McCluskey (Tabular) Method
systematically finds all prime implicants
Example Simplify (A,B,C,D) Sm(4,5,6,8,9,10,13)
Sd(0,7,15)
Implication Table Column I
0000 0100
1000 0101 0110
1001 1010
0111 1101 1111
Step 1 List minterms and dont cares using
their binary representation, and
group according to number of 1s
Principle Combine terms in adjacent groups which
differ in a single variable, and eliminate
the changing variable.
Remarks Only terms in adjacent groups have to
be compared with one another. Terms in
non-adjacent groups differ in more than one
variable. Thus can not be combined
15
Q M Method
(A,B,C,D) Sm(4,5,6,8,9,10,13) Sd(0,7,15)
Step 2 Compare elements of a
group with k 1's against those with
k1 1's. If they differ by one bit,
eliminate changing variable and
place reduced term in next
column. E.g., 0000 vs. 0100 yields
0-00 0000 vs. 1000 yields
-000
Implication Table Column I
Column II 0000 0-00
-000 0100
1000 010- 01-0
0101 100- 0110 10-0
1001 1010 01-1
-101 0111 011- 1101
1-01 1111 -111
11-1
When a term is used in a combination, mark that
term with a check. If cannot be combined, mark
term with a star. These are the prime
implicants.
16
Q M Method
Step 2 Continues Compare
elements of a group in Col. II with
k 1's against those with k1 1's.
If they differ by one bit, eliminate
changing variable and place reduced
term in next column. E.g., 010-
vs. 011- yields 01- - -101
vs. -111 yields -1-1
Implication Table Column I
Column II Column III 0000 0-00
01-- -000 0100
-1-1 1000
010- 01-0 0101
100- 0110 10-0 1001
1010 01-1 -101
0111 011- 1101 1-01
1111 -111
11-1
17
Q M - Prime Implicant Chart
x
x x
x
x x
5,7,13,15 (-1-1) 4,5,6,7(01--) 9,13(1-01) 8,10(
10-0) 8,9(10-0) 0,8(-000) 0,4(0-00)
5,7,13,15 (-1-1) 4,5,6,7(01--) 9,13(1-01) 8,10(
10-0) 8,9(10-0) 0,8(-000) 0,4(0-00)
x
x
x
x
x
x
x x
x x
x x x
x
x x x
x
x
x
4 5 6 8 9 10 13
4 5 6 8 9 10 13
rows prime implicants columns minterms place
an "X" if minterm is covered by the prime
implicant
If column has a single X, than the implicant
associated with that row is essential. It must
appear in minimum cover
18
Gate Logic

• Review

Switches


NOT
X
False
X
Description
Truth Table
Switches
Z 1 if X and Y
Y
X
Z
false
are both 1
0
0
0
AND
X Y
1
0
0
0
1
0
true
1
1
1
X Y
Switches
False

OR
19
Example Logic circuit its Switch realization
s a (dw)(c e)
Gate realization
Switch realization
20
Logic Functions NAND
Switches
NAND

False
Description
Truth Table
Switches
Z 1 if X and Y
Y
X
Z
false
are both 1
AND
0
0
0
X Y
1
0
0
0
1
0
true
1
1
1
X Y
Note that switch circuit of the NAND gate is the
complement of the switch circuit for the AND gate.
21
Logic Functions NAND, NOR, XOR, XNOR
Switches
NOR

False
Switches
False

OR
Note that switch circuit of the NOR gate is the
complement of the switch circuit for the OR gate.