Karnaugh map minimizationBasic Terms (1)

minterm0 0 0 0

0 1 1 1

1 1 1 11 0 1 1

maxterm

- a single 1

- a single 0

1.

implicant0 0 0 0

0 1 1 1

1 1 1 11 0 1 1

- any group of adjacent 1’s (or 0’s) of the size 1, 2, 4, 8, 16, ...

2.

prime implicant0 0 0 0

0 1 1 1

1 1 1 11 0 1 1

3.

an implicant that is not contained within any larger implicant

Karnaugh map minimizationBasic Terms (2)

1* - 1 (minterm) that is included in only one prime implicant

0 0 0 00 1* 1 1*

1 1 1 11* 0 1* 1

4.

essential prime implicant0 0 0 0

0 1* 1 1*

1 1 1 11* 0 1* 1

5.

secondary prime implicant0 0 0 0

0 1* 1 1*

1 1 1 11* 0 1* 1

6.

0* - 0 (maxterm) that is included in only one prime implicant

prime implicant thatcontains 1* (or 0*)

prime implicant thatdoes not containany 1* (or 0*)

Karnaugh map minimization

Algorithm (1)

1. Fill Karnaugh map based on the function description

2. Decide about your goal• if your goal is the minimum sum-of-products form you will be covering 1’s• if your goal is the minimum product-of-sums form you will be covering 0’s

3. Find ALL prime implicants (covering 1’s or 0’s depending on step 2)

4. Find ALL 1*s or 0*s (depending on step 2), i.e., 1’s (0’s) belonging to only one prime implicant

5. Find ALL essential prime implicants, i.e., prime implicants containing 1*s ( or 0*s )

6. Identify the remaining prime implicants as secondary prime implicants

7. Find ALL 1’s (0’s) not covered by essential prime implicants

8. Cover 1’s (0’s) found in step 7 using the minimum number of the largest secondary prime implicants

Karnaugh map minimization

Algorithm (2)

9. Write the minimized equation of the function F or F (depending on step 2) in the sum-of-products form

• ALL essential prime implicants first• secondary prime implicants SELECTED in step 8 next

10.

Document all your stepsYou should be able to verify each step

independently of other steps

If your goal is the product-of-sums formAND you have chosen to cover 0’s in step 2,apply the DeMorgan’s theorem to the equationobtained in step 9 to obtain the equation for F.