LOGIC DESIGN

PART 5 Karnaugh Map

Why Do You Need To Know About Karnaugh Maps?

• Karnaugh Maps are used for many small design problems. It's true that many larger designs are done using computer implementations of different algorithms. However designs with a small number of variables occur frequently in interface problems and that makes learning Karnaugh Maps worthwhile. In addition, if you study Karnaugh Maps you will gain a great deal of insight into digital logic circuits.

You will learn:• Draw the Karnaugh Map for the function. • Use the information from a Karnaugh Map to

determine the smallest sum-of-products function.

What Does a Karnaugh Map Look Like?

• A Karnaugh Map is a grid-like representation of a truth table.

• Another way of presenting a truth table• A Karnaugh map has zero and one entries

at different positions. • Each position in a grid corresponds to a

truth table entry. • An example is shown on the slide

How Can a Karnaugh Map Help?• In the case of the Karnaugh Map the

advantage is that the Karnaugh Map is designed to present the information in a way that allows easy grouping of terms that can be combined.

• Take all ones in pairs and group them

Let's examine the map again. • The term on the left in the gray area of

the map corresponds to: • The term on the right in the gray area

of the map corresponds to: • These two terms can be combined to

give

NOTE:• The beauty of the Karnaugh Map is that it has been cleverly

designed so that any two adjacent cells in the map differ by a change in one variable.

• It's always a change of one variable any time you cross a horizontal or vertical cell boundaries.

• Notice that the order of terms isn't random. Look across the top boundary of the Karnaugh Map. Terms go 00, 01, 11, 10. However, in a Karnaugh Map terms are not arranged in numerical sequence!

• That's done deliberately to ensure that crossing each horizontal or vertical cell boundary will reflect a change of only one variable. In the numerical sequence, the middle two terms, 01, and 10 differ by two variables!

Example 1

There is a small surprise in one grouping. The lower left and the lower right 1s actually form a group. They differ only in having B and its' inverse. Consequently they can be combined. You will have to imagine that the right end and the left end are connected.

Remember these basic rules

• In a Karnaugh Map of any size, crossing a vertical or horizontal cell boundary is a change of only one variable - no matter how many variables there are.

• Each single cell that contains a 1 represents a minterm in the function, and each minterm can be thought of as a "product" term with N variables.

• To combine variables, use groups of 2, 4, 8, etc. A group of 2 in an N-variable Karnaugh map will give you a "product" term with N-1 variables. A group of 4 will have N-2 variables, etc.

• You will never have a group of 3, a group of 5, etc. Don't even think about it. See the points above.