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Karnaugh map

Karnaugh map

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Karnaugh map

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  1. Karnaugh map Introduction Strategy for Minimization Minimization of Product-of-Sums Forms Minimization of More Complex Expressions Don't care Terms

  2. Introduction • Why karnaugh map • Example (With Boolean algebra) W = A + . B = A . ( B + ) + . B = A . B + A . + . B = A . ( B + ) + B ( A+ ) = A + B

  3. Introduction ( cont. ) • Using Boolean algebra for minimization causes it’s own problem because of it mainly being a trial and error process, and we can almost never be sure that we have reached a minimal representation. • If we can form a graphical notation for our Boolean algebra the insight need for the minimization will be less vital in solving the problems. We can come close to our aim by using a graphical notation named Karnaugh Map that will be defined in next slides

  4. As it can be seen, each box of the Karnaugh map corresponds to a row of the truth table and has been numbered accordingly Introduction ( cont. ) • Comparing Karnaugh Map and Boolean Algebra Truth Table Karnaugh Map A 0 1 B 0 1 W This form of representing w in the following example is called a Sum of Product (SOP) Which will be define in next slides W = . B + A . + A . B = W = . B + A . B + A . + A .B= W= B ( + A ) + A ( + B ) = A + B

  5. Strategy for Minimization • Terminology • Minimization Procedure

  6. Terminology • Implicant : Product term that implies function • Prime Implicant: An Implicant that is not completely covered by any other Implicant but itself • Essential prime Implicant: A prime Implicant that has a minter not covered by any other prime Implicant • Product term: An and expression

  7. Terminology • Minterm: We define a Minterm to be a product that contains all variables of that particular switching function in either complemented or non-complemented form • Maxterm: We define a Maxterm to be a sum that contains all variables of that particular switching function in either complemented or non-complemented form • Standard SOP(Sum Of Products): In standard SOP, the products are obtained directly from the Karnaugh map or truth table, so the SOP contains all of the variables of the function • Standard POS(Product Of Sums): In standard POS, the products are obtained directly from the Karnaugh map or truth table, so the POScontains all of the variables of the function

  8. Terminology ( cont. ) • A simpler shorthand form of representing a SOP is to use the number of the Minterms that appear in that representation. In the following example for instance we could have written Karnaugh Map AB 0 01 11 10 C 0 2 0 1 3 1 7 5 6 4 W =

  9. Terminology ( cont. ) • Sometimes writing an expression in a POS form is easier as seen in the following example: Karnaugh Map AB 00 01 11 10 C 0 2 1 0 3 1 5 7 6 4 W = w = (a + b + c) . (+ b + c)

  10. Strategy for Minimization • Terminology • Minimization Procedure