Simplification of incompletely specified functions using QM Tech.

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Presentation transcript:

Simplification of incompletely specified functions using QM Tech.

How to handle QM Tech when don’t care terms are present? In order to obtain a minimum solution when don’t care terms are present, we have to modify the QM Tech procedure. In the process of finding the prime implicants,we will treat don’t care term as if they were required minterms. In this way, they can be combined with other minterms to eliminate as many literals as possible

When forming the prime implicant chart, the don’t cares are not listed on the top In this way, when the prime implicant chart is solved, all of the required minterms will be covered by one of the prime implicants.

F(A,B,C,D)=∑m(2,3,7,9,11,13)+ ∑d(1,10,15) (1,3) (1,9) (2,3) (2,10) 0 0 – – (3,7) (3,11) (9,11) (9,13) (10,11) 0 – – 1 1 – – (7,15) (11,15) (13,15) – – 1 (1,3,9,11) (2,3,10,11) - 0 – – (3,7,11,15) (9,11,13,15)

Prime implicant chart The don’t care terms are omitted when forming this chart (1,3,9,11) * * * (2,3,10,11) * * * (3,7,11,15) * * * (9,11,13,15) * * *

(1,3,9,11) * * * #(2,3,10,11) * * * #(3,7,11,15) * * * #(9,11,13,15) * * * We can see that 2,7,13 minterms are covered only group marked as #, so these are essential prime implicants.

(1,3,9,11) * * * #(2,3,10,11) * * * #(3,7,11,15) * * * #(9,11,13,15) * * *

(1,3,9,11) * * * # (2,3,10,11) * * * # (3,7,11,15) * * * # (9,11,13,15) * * *

Reduced expression: F=B’C+CD+AD NOTE: In the process of simplification we have automatically assigned values to the don’t care in the original truth table for F. If we replace each term in the final expression for F by its corresponding sum of minterms, the result is F=(m2+m3+m10+m11) +(m3+m7+m11+m15)+(m9+m11+m13+m15) Since m10 and m15 appear in this expression and m1 does not, this implies that the don’t care in the original truth table for F have been assigned as follows: For ABCD=0001, F=0; For ABCD=1010, F=1; For ABCD=1111, F=1;

CLASS TEST F(A,B,C,D)=∑(0,1,2,4,6,9,11,13,15) and its having some don’t care terms. Find the don’t care terms if the reduced expression is F= AD+ CD’ + A’C’ Solution: d(5,10,14)