SYEN 3330 Digital Systems Jung H. Kim Chapter 2-5 1 SYEN 3330 Digital Systems Chapter 2 – Part 5.

SYEN 3330 Digital Systems Jung H. Kim Chapter 2-5 1

SYEN 3330 Digital Systems

Chapter 2 – Part 5

SYEN 3330 Digital Systems Chapter 2-5 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 teach other.


Three-Variable Maps

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

Y Z

X

1376 5

4

2

0


Three-Variable Maps

• Example Shapes of Rectangles:

0 1 3 2

5 64 7X

Y

Z

X

Y

Z Z


Three Variable Maps

K-Maps can be used to simplify Boolean functions by systematic methods. Terms are selected to cover the "ones" in the map.

Example: Simplify F(x,y,z) = m(1,2,3,5,7)

F(x,y,z) = x y+ z


Three-Variable Map Simplification

• F(X,Y,Z) = (0,1,2,4,6,7)


Four Variable Maps

x

y

z

w

w'x'y'z'w'x'y'z w'x'yz'w'x'yz'

w'xy'z' w'xy'z w'xyz'w'xyz

wx'y'z' wx'y'z wx'yz'wx'yz

wxy'z' wxy'z wxyz'wxyz

x

y

z

w

m0 m1 m2m3

m4 m5 m6m7

m8 m9 m10m11

m12 m13 m14m15


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")


8 9 1011

12 13 1415

Four-Variable Maps

• Example Shapes of Rectangles:

0 1 3 2

5 64 7

X

Y

Z

X

Y

Z Z

XW

W


Four-Variable Map Simplification

• F(W,X,Y,Z) = (0, 2,4,5,6,7,8,10,13,19)


Four-Variable Map Simplification

• F(W,X,Y,Z) = (3,4,5,7,13,14,15,17)


Systematic Simplification

•A Prime Implicant is a product term obtained by combining the maximum possible number of adjacent squares in the map.

•A is 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.


Example of Prime Implicants

Example: F(ABCD) = m(0,2,3,5,7,8,9,10,11,13,15)

Find ALL Prime Implicants

ESSENTIAL Prime Implicants


Prime Implicant Practice

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


Systematic Approach

(See Supplement 1)

(No Don’t Cares)

1. Select all Essential PI’s

2. Find and delete all Less Than PI’s

3. Repeat 1) and 2) until all minterms are covered

If Cycles Occur:

4. Arbitrarily select a PI and generate a cover.

5. Delete the selected PI and generate a new cover

6. Select the cover with fewer literals

7. If a new cycle appears, repeat steps 4), 5), and 6) and compare all solutions for the best.


Other PI SelectionOnce the Essential Prime Implicants are selected, we need to "prune" the solution set further. To do this, we determine which can be eliminated by finding Less Than PIs and Redundant PIs.

Less Than PIs:

PIi is said to be Less Than PIj if PIi contains at least as many literals as PIj and PIj covers at least all of the as yet uncovered minterms that PIi covers.

SecondaryEssential PIs:

Once the less than PIs are removed from consideration, new PIs become essential and they are called Secondary Essential PIs.

Redundant PIs:These are PIs whose minterms have been completely covered by the PIs selected and are removed from consideration.


Example 2 from Supplement 1

Select Essential PIs: Eliminate Less Than Pis:


Example 2 (Continued)

Select SecondaryEssential PIs:

Eliminate Redundant PIs:


Another Example

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


Five Variable or More K-Maps

For five variable problems, we use two adjacent K-Maps. It becomes harder to visualize adjacent minterms for selecting PIs. You can extend the problem to six variables by using four K-Maps.


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.


Example: BCD “5 or More”

A function F1(w,x,y,z) which is defined as "5 or greater" over a BCD input, as below. Don't cares are on non-BCD values.

F1(w,x,y,z) = w + x z + x y

This is slightly lower in cost than F2 where the don't cares are required to be "0".

F2(w,x,y,z) = w x z + w x y + wxy

For this particular function, note that the literal cost of the complement of F1(w,x,y,z), meaning "4 or less", is not changed by using the don't cares.


Product of Sums Example

• F(A,B,C,D) = (3,9,11,12,13,14,15) +

d (1,4,6)

• Use and take complement of result:F

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SYEN 3330 Digital Systems Jung H. Kim Chapter 2-5 1 SYEN 3330 Digital Systems Chapter 2 – Part 5.

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