Don't Care Minimization

Don't Care Minimization - Another example of using dont...

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Another example of using don’t cares to simplify logic equations: Suppose that we want to build a logic circuit that differentiates between multiples of 3 and non-multiples of 3. Additionally, suppose that we also know that all of the numbers will be input as 4-bit values, and that the circuit need only consider valid binary-coded decimal numbers. Assume that we are using the standard BCD representation, i.e., 8-4-2-1. First, let’s show the truth table representation for this circuit, if we ignore the fact that invalid BCD inputs need not be considered. In all cases, assume that A is the most significant variable, and that D is the least significant variable. A B C D f 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 1 1 1 0 1 0 0 0 0 1 0 1 0 0 1 1 0 1 0 1 1 1 0 1 0 0 0 0 1 0 0 1 1 1 0 1 0 0 1 0 1 1 0 1 1 0 0 1 1 1 0 1 0 1 1 1 0 0 1 1 1 1 1 In this case, the circuit outputs logic-1 for all multiples of 3, and it outputs logic-0 for all non-multiples of 3. We can show this information in a Karnaugh map: CD 00 01 11 10 AB 00 1 0 0 1 1 3 0 2 01 0 4 0 5 0 7 1 6 11 1 12 0 13 1 15 0 14 10 0 8 1 9 0 11 0 10 Since there are no logically adjacent ones in the map, we must represent all of the 1s as separate groups: f(A,B,C,D) = A’B’C’D’ + A’B’CD + A’BCD’ + AB’C’D + ABC’D’ + ABCD This is not a compact equation to say the least.

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As a first refinement, suppose that we now observe the fact that invalid BCD inputs need not be considered. For the sake of simplicity, we could say that every invalid BCD input will give an output of logic-0. This is “okay” because if the invalid BCD inputs will never be applied as inputs, it doesn’t matter that they all give outputs of 0. A
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Don't Care Minimization - Another example of using dont...

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