Logic Design - Chapter 4: Karnaugh Maps

CHAPTER 4

THE KARNAUGH-VEITCH- MAP

Contents

THE THREE VARIABLE KARNAUGH MAP THE FOUR VARIABLE KARNAUGH MAP Karnaugh Map Simplification of SOP Expressions KARNAUGH MAP PRODUCT OF SUM (POS)

SIMPLIFICATION 5-VARIABLE K-MAPS DON'T CARE CONDITIONS

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THE THREE VARIABLE KARNAUGH MAP

Karnaugh Maps

THE FOUR VARIABLE KARNAUGH MAP

Karnaugh Maps

Karnaugh Map Simplification of SOP ExpressionsGrouping the 1s 1. A group must contain either 1, 2, 4, 8,

or 16 cells. 2. Each cell in a group must be adjacent to

one or more cells in that same group. 3. Always include the largest possible

number of 1s in a group.4. Each 1 on the map must be included in at

least one group.

Karnaugh Maps

Determining the Minimum SOP Expression from the Map 1. Group the cells that have 1s. 2. Determine the minimum product terms for each group. For a 3-variable map:

(1) A 1-cell group yields a 3-variable product term (2) A 2-cell group yields a 2-variable product term (3) A 4-cell group yields a 1-variable term (4) An 8-cell group yields a value of 1 for the expression

For a 4-variable map ?

For a 5-variable map ?

3. When all the minimum product terms are derived from the Karnaugh map, they are summed to form the minimum SOP expression.

Karnaugh Maps

EXAMPLE Simplify the Boolean expression:

F(x,y,z) = Σ (0,1,6,7)

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EXAMPLE: Simplify the Boolean expression:

F(x,y,z) = Σ (0,2,5,7)

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EXAMPLE: Group the 1's in each of the following

Karnaugh maps:

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EXAMPLE:

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KARNAUGH MAP PRODUCT OF SUM (POS) SIMPLIFICATION If we mark the empty squares by 0's and

combine them into valid adjacent squares, we obtain a simplified expression of the complement of the function, i.e., of F'.

The complement of F' gives us back the function F.

Because of Demorgan's theorem, the function so obtained is automatically in the product of sums form.

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EXAMPLE Simplify the following Boolean function in (a) sum

of products and (b) product of sums. F(w,x,y,z) = Σ (0,1,2,3,10,11,14)

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EXAMPLE Use a Karnaugh map to

minimize the following POS expression.

(x+y+z)(w+x+y'+z) (w'+x+y+z') (w+x'+y+z) (w'+x'+y+z)

(w+x+y+z)(w'+x+y+z)(w+x+y'+z)(w'+x+y+z')(w+x'+y+z)(w'+x'+y+z)

=Π(0,8,2,9,4,12)

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5-VARIABLE K-MAPS (1/2)

K-maps of more than 4 variables are more difficult to use because the geometry (hypercube configurations) for combining adjacent squares becomes more involved.

For 5 variables, e.g. v, w, x, y, z, we need 25 = 32 squares.

Each square has 5 neighbours.

ELC224C Karnaugh Maps

5-VARIABLE K-MAPS (2/2) Organized as two 4-variable K-maps. One for v'

and the other for v.

m20 m21

w

y

m23 m22

m16 m17 m19 m1800

01

11

10

00 01 11 10

z

wxyz

m28 m29 m31 m30

m24 m25 m27 m26

x

m4 m5

w

y

m7 m6

m0 m1 m3 m200

01

11

10

00 01 11 10

z

wxyz

m12 m13 m15 m14

m8 m9 m11 m10

x

v ' v

Corresponding squares of each map are adjacent.Can visualise this as one 4-variable K-map being on

TOP of the other 4-variable K-map.

ELC224C Karnaugh Maps

DON'T CARE CONDITIONS Sometimes a situation arises in which some input

variable combinations are not allowed. Since these unallowed states will never occur in an

application involving the BCD code, they can be treated as "don't care" terms

for each "don't care" term, an X is placed in the cell.

When grouping the 1's, Xs can be treated as 1's to make a larger grouping or as 0s if they cannot be used to advantage.

Be careful do not make a group entirely of x's.

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Example

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Inputs Output x y y w Y 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 1 0 0 1 0 0 0 0 1 0 1 0 0 1 1 0 0 0 1 1 1 1 1 0 0 0 1 1 0 0 1 1 1 0 1 0 X 1 0 1 1 X 1 1 0 0 X 1 1 0 1 X 1 1 1 0 X 1 1 1 1 X

Example Simplify the following

Boolean function F, where d represents the set of do not care conditions

F(w,x,y,z) = Σ (0,1,2,8,10,11)

d(w,x,y,z) = Σ (4,6,12,13)

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