It happens because one of the AND gates is holding the output 1 before the
transition and the other AND gate holds it 1 after the transition. The spike occurs if
it ...
Karnaugh Maps Objectives This section presents a technique for simplifying logical expressions. It will: Define Karnaugh and establish the correspondence between Karnaugh maps and truth tables and logical expressions. Show how to use Karnaugh maps to derive minimal sumof-products and product-of-sums expressions. Introduce the concept of "don't care" entries and show how to extend Karnaugh map techniques to include maps with don't care entries.
Reading Assignment
Sections 2.6 and 2.7 from the text
Elec 326
1
Karnaugh Maps
Karnaugh Map Definitions A Karnaugh map is a two-dimensional truth-table. Unlike ordinary (i.e., one-dimensional) truth tables, however, certain logical network simplifications can be easily recognized from a Karnaugh map. A B Z 0 0 0 0 1 1 1 0 1 1 1 0 Truth Table
0
1
Z A 0 B 0 0
1
0
1
Z
A
B
1
1 1 0
Type 2 Map
Type 1 Map
Two-Variable Maps AB C Z 0 00 0 0 01 1 0 10 1 0 11 0 1 00 1 1 01 0 1 10 0 1 11 1 Truth Table
Z
A
B
0
1
1
0
0
1
1
0
Z A B,C 0 00 0
1 1
01
1
0
11
0
1
10
1
0
C
Type 1 Map
Type 2 Map
Three-Variable Maps Elec 326
2
Karnaugh Maps
1
A 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1
B 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1
C 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1
Z 0 0 0 1 0 1 1 1 0 1 1 1 1 1 1 1
D 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
Z
Z A,B C,D 00 01 11 10 00 0 0 1 0
A
C
0
0
1
0
0
1
1
1
1
1
1
1
0
1
1
1
D
01 0
1
1
1
11 1
1
1
1
10 0
1
1
1
B Type 1 Map
Type 2 Map
Four-Variable Maps
Truth Table
Elec 326
3
Karnaugh Maps
The interpretation of a type 1 map is that the rows or columns labeled with a variable correspond to region of the map where that variable has value 1. A
A D
C B A=1 region
A D
C
D
C
B B=1 region
A D
C
B C=1 region
B D=1 region
Numbering of Karnaugh Map Squares.
Elec 326
Z A 0 B 0 0
1 2
1
3
1
Z A B,C 0 00 0
1 4
Z A,B C,D 00 01 11 10 00 0 4 12 8
01
1
5
01 1
5
13 9
11
3
7
11 3
7
15 11
10
2
6
10 2
6 14 10
4
Karnaugh Maps
2
Exercise: Plot the following expression on a Karnaugh map. Z = (A•B)⊕(C+D) A Z = A•B•(C+D)' + (A•B)'•(C+D)
B
= A•B•C'•D' + (A'+B')•(C+D) = A•B•C'•D' + A'•C + A'•D + B'•C + B'•D
1
0
1
0
0
0
1
1
1
1
1
1
0
1
1
0
D
C
Elec 326
5
Karnaugh Maps
Minimal Sum-Of-Products Expressions Ordering of Squares
The important feature of the ordering of squares is that the squares are numbered so that the binary representations for the numbers of two adjacent squares differ in exactly one position. This is due to the use of a Gray code (one in which adjacent
numbers differ in only one position) to label the edges of a type 2 map.
The labels for the type 1 map must be chosen to guarantee this
property.
Elec 326
Note that squares at opposite ends of the same row or column also have this property (i.e., their associated numbers differ in exactly one position).
6
Karnaugh Maps
3
Merging Adjacent Product Terms Z Z
C
A 0
0
0
0
0
1
1
0
0
0
0
0
0
0
0
0
= (A'+A)•B•C'•D = 1•B•C'•D
D
= B•C'•D A' B C' D A B C' D
B
= m5 + m13 = A'•B•C'•D + A•B•C'•D
Z
B C' D
Z
Example A • B • C'
Z
A
A' • C • D C
0
0
1
0
0
0
1
0
1
1
0
0
0
0
0
0
D
B
Elec 326
7
Karnaugh Maps
For k-variable maps, this reduction technique can also be applied to groupings of 4,8,16,...,2k rectangles all of whose binary numbers agree in (k-2),(k-3),(k4),...,0 positions, respectively. Z
C
A 0
0
0
0
0
1
1
0
0
1
1
0
0
0
0
0
D
B
Z= = = = = =
Elec 326
m5 + m7 + m13 + m15 A'•B•C'•D + A'•B•C•D + A•B•C'•D + A•B•C•D (A'•C' + A'•C + A•C' + A•C) • (B•D) (A'•(C' + C) + A•(C' + C)) • (B•D) (A' + A) • (B•D) B•D
8
Karnaugh Maps
4
Basic Karnaugh Map Groupings for Three-Variable Maps. Z1
Z3
A
Z5
A
1
1
C B
Z2
1
C
B
B
1
Z4
A
A 1
1
C
1 1
Z6
A
A 1
1 B
C
1
B
Elec 326
1
1
1
1
1
C
C B
1
1
9
Karnaugh Maps
Basic Karnaugh Map Groupings for Four-Variable Maps. Z7
A
Z8
Z9
A
A
Z10
A
1 1
1 C
C
A
1
1
1
Elec 326
Z12
D
D
C
C
1
1
B A 1
1
1
A 1
Z14
1 D
C
C
1
1 B
10
D
A 1
D
B
1
B
Z13
1
C B
D
B
B Z11
1
1
D C
1
1 D
1
1 B
Karnaugh Maps
5
Z15
Z16
A
Z17
A
A
1 1
D
1
C
1
1
1
1
C
1
D C
B
1
1
1
1
1
1
1
1
1
1
D
D
C
C
C B
Elec 326
1
1
1 B
11
1
1
1
D
1
Z20
A 1
1
B
Z19
A 1
1
1
B
Z18
1
1
A 1
1
1
1
1
1
1
D
1 B
Karnaugh Maps
Rules for Grouping:
The number of squares in a grouping is 2i for some i such that 1 ≤ i ≤ k. There are exactly k-i variables that have constant value for all squares in the grouping.
Resulting Product Terms:
Elec 326
If X is a variable that has value 0 in all of the squares in the grouping, then the literal X' is in the product term. If X is a variable that has value 1 in all of the squares in the grouping, then the literal X is in the product term. If X is a variable that has value 0 for some squares in the grouping and value 1 for others, then neither X' nor X are in the product term. 12
Karnaugh Maps
6
Invalid Karnaugh Map Groupings.
Z
C
A
loop 1
Z
0
1
0
0
0
1
1
0
1
1
1
0
0
1
1
0
D C
A
1
0
0
1
1
1
0
1
1
1
1
1
0
1
1
1
B
loop 2 D
B
Violates Rule 1
Violates Rule 2
Elec 326
13
Karnaugh Maps
In order to minimize the resulting logical expression, the groupings should be selected as follows:
Identify those groupings that are maximal in the sense that they are not contained in any other possible grouping. The product terms obtained from such groupings are called prime implicants. A distinguished 1-cell is a cell that is covered by only one prime
implicant.
An essential prime implicant is one that covers a distiquished 1-
cell.
Use the fewest possible number of maximal groupings needed to cover all of the squares marked with a 1.
Examples:
#1 C #4
Elec 326
#2
Z
A
1
0
1
1
0
1
1
0
0
1
1
1
1
0
1
1
B 14
D
#3 Karnaugh Maps
7
Z
#4 C
#3
#2
A
1
1
0
1
0
1
1
1
1
1
1
0
1
0
0
0
#5
Z
#3
#4 D C #1
A 1
0
1
0
1
1
1
1
1
1
0
1
0
0
0
#5
B
a. Non-minimal Product Terms
Z
#2
1
#3
D C #1
A 1
0
1
0
1
1
1
1
1
1
0
1
0
0
0
#4
B
#2
1
D #1
B
b. Non-minimal Selection of Prime Implicants
c. Minimal Grouping
a. Z = A•B•D+A•B'•C'+A'•B•C'+A'•C•D+A'•B'•D' b. Z = B•D+A•B'•C'+A'•B•C'+A'•C•D+A'•B'•D' c. Z = B•D+A•B'•C'+A'•C'•D'+A'•B'•C W3
C
W3
A
A
1
0
1
1
1
0
1
1
1
1
1
1
1
1
1
1
0
1
1
0
0
1
1
0
1
1
1
0
1
1
1
0
D C
B
D
B
a. W3 = C'D+AC'+BC+A'B'D' b. W3 = BD+AB+B'C'+A'CD' Elec 326
15
Karnaugh Maps
Exercise: Derive minimal sum-of-products logical expressions from the following Karnaugh maps. X
Y
A
1
1
1
0
1
1
0
0
A
B
1
1
1
0
1
1
0
0
0
0
1
0
0
0
1
1
D
0
0
1
0
0
0
1
1
D C
C
B
X = A•B + A'•B'•D' + A'•B'•C + B•C•D' Y = A'•C' + A•C•D' + A•B•C + B•C'•D'
Elec 326
16
Karnaugh Maps
8
Minimal Product-Of-Sums Expressions Merging Adjacent Product Terms. Z
B
A 1
1
1
1
0
0
1
1
C
A' B' C' A B' C'
Z
B' C'
Z
OR-AND Networks
Karnaugh Map
M3•M7 = (A+B'+C')•(A'+B'+C') = (A•A')+(B'+C') = 0 + (B'+C') = B' + C'
Elec 326
17
Karnaugh Maps
Rules for Grouping:
Same as for sum-of-products, except that zero's are grouped instead of ones.
Resulting Sum Terms:
If variable X has value 0 for all squares in the group, then the literal X is in the sum term. If variable X has value 1 for all squares in the group, then the literal X' is in the sum term. If variable X has value 0 for some squares in the group and value 1 for the others, then that variable does not appear in the sum term.
Prime Implicate:
Elec 326
Maximal grouping of zeros.
18
Karnaugh Maps
9
Examples: Z B'+C+D A+D'
C
A
1
0
0
1
0
0
1
1
0
0
1
1
1
1
1
0
D
A'+B+C'+D
B
Z = (A+D')•(B'+C+D)•(A'+B+C'+D) Z1
C
Z2
A
A
1
0
1
1
1
0
1
1
0
1
1
0
0
0
0
0
0
1
1
1
0
0
0
0
1
0
1
1
1
0
0
1
D C
B
D
B
Z1 = (A+B'+D)•(B+C+D')•(A+B+D') Z2 = D'•(A+B')•(B'+C') Elec 326
19
Karnaugh Maps
Comparison of Sum-of-Products and Product-ofSums Expressions. Z1
C
Z1
A 1
0
1
0
0
0
0
0
0
0
0
1
1
0
1
1
D C
A 1
0
1
0
0
0
0
0
0
0
0
1
1
0
1
1
D
B
B
Z1 = (A+B')•(A+D')•(B'+D')•(A'+B+C) Z1 = (A•B•D') + (A•B'•C) + (A'•B'•D')
Elec 326
20
Karnaugh Maps
10
Exercise: Derive minimal SOP and POS expressions from the following Karnaugh maps. X
X
A 1
1
1
1
1
0
0
1
A 1
1
1
1
1
0
0
1
1
0
0
0
1
1
0
0
B D 1
0
0
0
1
1
0
0
D C
C
B
X = B•D' + A•C' + A•D' + B•C'
Y = (B' + D') • (A' + C')
Elec 326
21
Karnaugh Maps
Don't Care Entries Assignment of Values to Don't Care Entries. Z
B
Z
A
Z
A
Z
A
A
1
1
1
1
1
1
1
1
d
1
1
1
0
1
1
1
d
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
C B
C B
C B
Z = B'+A'•C Z = A•B'+B'•C' Z
C
A 0
d
1
0
0
d
0
0
1
d
d
0
1
d
D C
d
B
A 1
d
d
1
0
0
d
0
1
1
0
d
1
d
d
1
D
B
Z = B•C + A•D'
Elec 326
Z = B'
Z
0
C
Z = (C+D')•(A'+B')
22
Karnaugh Maps
11
Minterm and Maxterm Lists with Don't Care Entries. ΣA,B,C(0,d1,d3,4,5) and ΠA,B,C(d1,2,d3,6,7) ΣA,B,C(6,7,8,d10,d11,d12,d13,d14,d15) ΠA,B,C(0,1,2,3,4,5,9,d10,d11,d12,d13,d14,d15)
Elec 326
23
Karnaugh Maps
Five- and Six-Variable Maps Five-Variable Maps.
0 1 D
3 2
4
12
B 8
5
7 6
13 9 E 15 11 14 10
Z
C
B 16 20 28 24 17 21 29 25 E A D 19 23 31 27 18 22 30 26
D
B 0
4
12 8
1
5
13 9
3
7
15 11
2
6
14 10 C
B 16 20 28 24 E D
17 21 29 25
E
19 23 31 27 18 22 30 26 C A
C
Five-Variable Map Structure
Elec 326
Alternate Version of Five-Variable Map
24
Karnaugh Maps
12
Six-Variable Maps C 12 8 1 5 13 9 F 3 7 15 11 E 2 6 14 10 0
4
Z
C
D
C 16 20 28 24 17 21 29 25 F E 19 23 31 27 18 22 30 26
E
B
4
12 8
1
5
13 9
3
7
15 11
2
6
16 20 28 24 17 21 29 25
F E
14 10 D
D
C
0
18 22 30 26 D
C
33 37 45 41 A
49 53 61 57
F
35 39 47 43
E
C
48 52 60 56
32 36 44 40
C 48 52 60 56 49 53 61 57 F E 51 55 63 59 50 54 62 58
F
19 23 31 27
34 38 46 42
F A
51 55 63 59
E
50 54 62 58 D
D
D
B
C 32 36 44 40 33 37 45 41 F E 35 39 47 43 34 38 46 42
Alternate Version of Six-Variable Map
D
Six Variable Map Structure Elec 326
25
Karnaugh Maps
Examples. Z
B
B' •C' • E' D
B
1
0
0
0
1
0
0
0
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
1
0
0
0
1
0
0
0
C
E B•E
C A
Z = B'•C'•E'+B•E Z
D
B
B
0
0
1
1
0
0
1
1
0
1
1
0
0
1
1
0
1
1
1
0
0
1
1
1
1
1
0
0
0
0
1
1
C
E
C A
A'B'D + CE + ABD + BD'E'
Elec 326
26
Karnaugh Maps
13
Z
B
D
B
0
0
1
0
0
0
0
0
0
1
1
1
0
1
1
1
0
1
1
1
0
1
1
1
0
0
1
0
0
0
1
1
C
E
C A
Z = (B+C)•(B+E•(A'+D+E)•(A+C+E) Z
B
D
B
0
d
1
0
d
d
d
d
0
d
1
0
0
0
1
0
d
d
1
1
d
d
d
d
d
d
1
1
1
1
1
1
C
E
C A
Z = B•C + D
Elec 326
27
Karnaugh Maps
Timing Hazards Exercise: Draw a timing diagram for the following circuit assuming an equal unit delay for each gate. X
A B
Y
A B X Y
Elec 326
28
Karnaugh Maps
14
Static 1 hazards A=1 S
P NS
Z
Q
B=1 S P NS Q Z
0 spike, static 1 hazard
Without the delay in the gates, the output Z would be constant at 1. The spike is due to the gate delay and the fact that not every path from on input to the output is the same length.
Elec 326
29
Karnaugh Maps
There is a simple way to prevent the hazards using Karnaugh maps. A Z S
0
1
1
0
0
0
1
1
B
Elec 326
The hazard occurs when the output is 1 before and after the transition. It happens because one of the AND gates is holding the output 1 before the transition and the other AND gate holds it 1 after the transition. The spike occurs if it is possible for the first to turn off before the second turns on. This can happen anytime there are to adjacent squares on the Karnaugh map that are both 1 and not both covered by a common loop. The hazard occurs in making a transition from one square to the other. The way to prevent it is to put in an extra product term that is 1 on both sides of the transition. On the Karnaugh map this is a loop covering the two adjacent one squares. 30
Karnaugh Maps
15
The loop added to prevent the hazard is redundant and not needed to realize the logical expression. Its only purpose is to prevent the hazard. A S Z
B
This technique will avoid all static 1 hazards if only one variable is changed at a time. It is not possible to get a 1 spike from a 2-level AND-OR network by changing only one variable. Why?
Elec 326
31
Karnaugh Maps
Static 0 hazard S A=0 S
P
P NS
Z
Q
Q
B=0
1 spike, static 0 hazard
Z
The static 0 hazard is caused by two OR gates where one holds the output zero before the transition and the other after the transition, and it is possible for them to both be 1 during the transition The fix is to put in an extra OR gate to hold the output zero during the transition
Z
S
A S
A 0
0
1
1
0
1
1
0
Z
B
B
Elec 326
32
Karnaugh Maps
16
Hazards can frequently be ignored. If the signal Z above is not used during the transition, then we don't care if it has a spike. When a signal is used as the input to a flip-flop, it is only sampled with the clock makes a transition. Therefore, if we can make sure that any spikes occur at some other time, they will not be a problem.
Elec 326
33
Karnaugh Maps
17
