Jung H. Kim Chapter 25 1

1
SYEN 3330 Digital Systems

• Chapter 2 Part 5

2
Three-Variable Maps

• Reduced literal product terms for SOP standard
  forms correspond to rectangles on K-maps
  containing cell counts that are powers of 2.
• Rectangles of 2 cells represent 2 adjacent
  minterms of 4 cells represent 4 minterms that
  form a pairwise adjacent ring.
• Rectangles can be in many different positions on
  the K-map since adjacencies are not confined to
  cells truly next to each other.

3
Three-Variable Maps

• Topological warps of 3-variable K-maps that show
  all adjacencies
• Venn Diagram ? Cylinder

0
4
6
5
7
3
1
2
4
Three-Variable Maps

• Example Shapes of Rectangles

1
2
3
0
5
7
6
4
5
Three Variable Maps
F(x,y,z) ?x y z
6
Three-Variable Map Simplification

• F(X,Y,Z) ?(0,1,2,4,6,7)

7
Four Variable Maps
8
Four Variable Terms

• Four variable maps can have terms of
• Single one 4 variables, (i.e. Minterm)
• Two ones 3 variables,
• Four ones 2 variables
• Eight ones 1 variable,
• Sixteen ones zero variables (i.e. Constant "1")

9
Four-Variable Maps

• Example Shapes of Rectangles

Y
?Y
1
2
3
0
?X
?W
6
5
4
7
X
12
14
15
13
W
10
11
8
9
?X
?Z
?Z
Z
10
Four-Variable Map Simplification

• F(W,X,Y,Z) ?(0, 2,4,5,6,7,8,10,13)

11
Four-Variable Map Simplification

• F(W,X,Y,Z) ?(3,4,5,7,13,14,15)

12
Systematic Simplification

• A Prime Implicant is a product term obtained by
  combining the maximum possible number of adjacent
  squares in the map.
• A prime implicant is called an Essential Prime
  Implicant if it is the only prime implicant that
  covers (includes) one or more minterms.
• Prime Implicants and Essential Prime Implicants
  can be determined by inspection of the K-Map.
• A set of prime implicants that "covers all
  minterms" means that, for each minterm of the
  function, there is at least one prime implicant
  in the selected set of prime implicants that
  includes the minterm.

13
Example of Prime Implicants
14
Prime Implicant Practice

• F(A,B,C,D) ?(0,2,3,8,9,10,11,12,13,14,15)

15
Systematic Approach
(No Dont Cares)

• Select all Essential PIs
• Find and delete all Less Than PIs
• Repeat 1) and 2) until all minterms are covered
• If Cycles Occur
• Arbitrarily select a PI and generate a cover.
• Delete the selected PI and generate a new cover
• Select the cover with fewer literals
• If a new cycle appears, repeat steps 4), 5), and
  6) and compare all solutions for the best.

16
Other PI Selection
17
Example 2 from Supplement 1
18
Example 2 (Continued)
19
Another Example

• G(A,B,C,D) ? (0,2,3,4,7,12,13,14,15)

20
Five Variable or More K-Maps
21
Don't Cares in K-Maps

• Sometimes a function table contains entries for
  which it is known the input values will never
  occur. In these cases, the output value need
  not be defined. By placing a don't care in
  the function table, it may be possible to arrive
  at a lower cost logic circuit.
• Don't cares are usually denoted with an "x" in
  the K-Map or function table.
• Example of Don't Cares - A logic function
  defined on 4-bit variables encoded as BCD digits
  where the four-bit input variables never exceed
  9, base 2. Symbols 1010, 1011, 1100, 1101, 1110,
  and 1111 will never occur. Thus, we DON'T CARE
  what the function value is for these
  combinations.
• Don't caresare used in minimization procedures
  in such a way that they may ultimately take on
  either a 0 or 1 value in the result.

22
Example BCD 5 or More
23
Product of Sums Example

• F(A,B,C,D) ? (3,9,11,12,13,14,15) ?d
  (1,4,6)