Karnaugh Maps
- This is from the supplimental material (starting 130 at the end
of the chapter)
- This is how we come up with a function from a truth table, even if we don't see it.
- This method works for up to four inputs
- This method relies on minterms.
- These are the terms we write down in a truth table as input
- Each includes the variable, or it's complement only once
- Each includes all of the variables.
- For 2 inputs the minterms are xy, x'y, xy' and x'y'
- In the truth table, the expression can be read directly
| X | Y | Minterm |
|---|
| 0 | 0 | X'Y' |
| 0 | 1 | X'Y |
| 1 | 0 | XY' |
| 1 | 1 | XY |
- Do a three input table.
- Do a four input table if we need it.
- All truth tables (and thus all binary functions) can be expressed
as an or of minterms
- Do this for exclusive or.
- Do this for sum in a full adder.
- Do this for carry out in a full adder.
- Which means, given a truth table, we can write down the formula for
this table.
- But these may not be simplified.
- We can do Boolean Algebra (Yuck)
- Or use a nuber of simplification methods, one known as Karnaugh maps, or Kmaps
- To do this we need a table of minterms.
- We then fill in this table with
- A 1 for each minterm which corresponds to a 1
- A blank (or 0) for minterms corresponding to a 0
- We then apply the following rules to simplify the expression the
kmap represents. (page 138)
- Form groups containing only 1s
- Only adjacient cells can be grouped, not diagionally
- The number of 1s in a group must be a power of two
- The groups must be as large as possible while still following all other rules
- All 1s must beloong to a group, even if it is a group of 1
- Overlapping groups are allowed
- Wrap around is allowed.
- Use the fewest number of groups possible
- Try the following
| w | x | y | z | f |
| 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 | 1 |
| 0 | 1 | 1 | 1 | 1 |
| 1 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 1 | 1 |
| 1 | 0 | 1 | 0 | 0 |
| 1 | 0 | 1 | 1 | 0 |
| 1 | 1 | 0 | 0 | 0 |
| 1 | 1 | 0 | 1 | 0 |
| 1 | 1 | 1 | 0 | 1 |
| 1 | 1 | 1 | 1 | 1 |