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SYEN 3330 Digital Systems. Chapter 2 – Part 5. 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.

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### SYEN 3330 Digital Systems

Chapter 2 – Part 5

SYEN 3330 Digital Systems

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.

SYEN 3330 Digital Systems

X

Y

Z

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

SYEN 3330 Digital Systems

Y

Y

X

X

Z

Z

Z

Three-Variable Maps
• Example Shapes of Rectangles:

1

2

3

0

5

7

6

4

SYEN 3330 Digital Systems

Three Variable Maps

F(x,y,z) = x y+ z

SYEN 3330 Digital Systems

Three-Variable Map Simplification
• F(X,Y,Z) = (0,1,2,4,6,7)

SYEN 3330 Digital Systems

y

y

m0

m1

m3

m2

w\'x\'y\'z\'

w\'x\'y\'z

w\'x\'yz\'

w\'x\'yz\'

m4

m5

m7

m6

x

w\'xy\'z\'

w\'xy\'z

w\'xyz

w\'xyz\'

m12

m13

m15

m14

x

w

wxy\'z\'

wxy\'z

wxyz

wxyz\'

m8

m9

m11

m10

w

wx\'y\'z\'

wx\'y\'z

wx\'yz

wx\'yz\'

z

z

Four Variable Maps

SYEN 3330 Digital Systems

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")

SYEN 3330 Digital Systems

Four-Variable Maps
• Example Shapes of Rectangles:

Y

Y

1

2

3

0

X

W

6

5

7

4

X

12

15

14

13

W

11

10

8

9

X

Z

Z

Z

SYEN 3330 Digital Systems

Four-Variable Map Simplification
• F(W,X,Y,Z) = (0, 2,4,5,6,7,8,10,13)

SYEN 3330 Digital Systems

Four-Variable Map Simplification
• F(W,X,Y,Z) = (3,4,5,7,13,14,15)

SYEN 3330 Digital Systems

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.

SYEN 3330 Digital Systems

Example of Prime Implicants

SYEN 3330 Digital Systems

Prime Implicant Practice
• F(A,B,C,D) = (0,2,3,8,9,10,11,12,13,14,15)

SYEN 3330 Digital Systems

(No Don’t Cares)

Systematic Approach
• Select all Essential PI’s
• Find and delete all Less Than PI’s
• 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.

SYEN 3330 Digital Systems

Other PI Selection

SYEN 3330 Digital Systems

Example 2 from Supplement 1

SYEN 3330 Digital Systems

Example 2 (Continued)

SYEN 3330 Digital Systems

Another Example
• G(A,B,C,D) =  (0,2,3,4,7,12,13,14,15)

SYEN 3330 Digital Systems

Five Variable or More K-Maps

SYEN 3330 Digital Systems

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 cares“are used in minimization procedures in such a way that they may ultimately take on either a 0 or 1 value in the result.

SYEN 3330 Digital Systems

Example: BCD “5 or More”

SYEN 3330 Digital Systems

Product of Sums Example
• F(A,B,C,D) =  (3,9,11,12,13,14,15) + d(1,4,6)

SYEN 3330 Digital Systems