3.3 Hardware Karnaugh Maps

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1 2P P = P = 3.3 Hardware UIntroduction A Karnaugh map is a graphical method of Boolean logic expression reduction. A Boolean expression can be reduced to its simplest form through the 4 simple steps involved in Karnaugh mapping. The Map can be used for expressions with 2, 3, 4, and 5 variable expressions/inputs A benefit of the Karnaugh map is that you don t need to memorize so many rules (like in Boolean Algebra) The number of cells in a Karnaugh map is equal to the total number of possible input variable combinations. 3 For three variables, the number of cells in the K map would be 2P 8 4 For four variables, the number of cells in the K map would be 2P 16 Hence we can say that the number of cells for n variable expression can be found by n UThe Format? The K map, like a truth table, is a means for showing the relationship between logic input and the desired output. The K map squares are labeled so that horizontally adjacent squares differ by only one variable. Note that each square in the top row is considered to be adjacent to a corresponding square in the bottom row. You can think of the top of the map as being wrapped around to touch the bottom of the map. Similarly, squares in the rightmost column with the leftmost squares. This is known as the wrap around property of K maps. Keep in mind that a cell is not adjacent to the cells that diagonally touch any of its corners In order for vertically and horizontally adjacent squares to differ by only one variable, the top-to-bottom labeling must be done in the order shown below. Page 1 of 9

2 For a standard expression, a 1 is placed on the Karnaugh map for each matching term The truth table gives the value of output X for each combination of input values. The K map gives the same information in a different format. Each case in the truth table corresponds to a square in the K map. When an expression is completely mapped, there will be a number of 1 s on the K map equal to the number of terms in the expression. The cells that do not have a 1 are the cells for which the expression is 0. Once a K map has been filled with 0 s and 1 s, the new expression can be obtained by ORing (or ADDing depending on the question) together those squares that contain a 1. Page 2 of 9

3 U2 Variable K-map: U3 Variable K-map: U4 Variable K-maps: Page 3 of 9

4 UGrouping/Looping: The next step to obtain the minimized expression after the expression has been mapped is combining those squares in the K map which contain 1 s. This process is called grouping/looping. At the time of grouping, a few rules must be kept in mind. U2-CELL GROUPING: A group must contain 1, 2, 4, 8, or 16 cells, which are all powers of 2. In the case of a 3 3-variable K map, 2P P= 8 cells is the biggest group size you can make. Each cell in a group must be adjacent to one or more cells in the same group, but all cells in the group do not have to be adjacent to each other. Always look for the largest possible group you can make in the K-map Each 1 on the map must be included in at least one group. The 1 s already in a group can be included in another group as long as the overlapping groups include noncommon 1s Page 4 of 9

5 U4-CELL GROUPING: U8-CELL GROUPING: Page 5 of 9

6 UAn example of grouping UAnother example Page 6 of 9

7 Construct a K-map for the following Truth-table and simplify Construct a K-map for the following Truth-table and simplify Page 7 of 9

8 There may well be more than one solution of equal complexity. Here is an example of a K map. There are 2 groups that are obvious. (one in orange, and one is light blue) In this example, the 2 terms formed are: W.Y.Z W.X.Y There is still one entry to account for. There is a 1 that can be joined to either of the 2 other entries to form a group. There is no best way to go about on this. Either way will take the same number of gates, inputs, etc. The last group could be grouped as: Page 8 of 9

9 A Boolean expression can be reduced to its simplest form through 4 simple steps involved in Karnaugh mapping. 1. Populate the K map with 1s as they relate to the truth table. 2. Group the adjacent 1s. 3. Analyze the groups. 4. List the reduced Boolean expression. Page 9 of 9

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