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9/15/09 - L8 Map ManupulationCopyright 2009 - Joanne DeGroat, ECE, OSU1 Map Manupulation Optimization, terms and don’t cares

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9/15/09 - L8 Map Manupulation Copyright 2009 - Joanne DeGroat, ECE, OSU2 Class 8 outline Prime Implicants Essential Prime implicants Non-essential prime implicants Product-of-Sums optimization Don’t cares Material from section 2-5 of text

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Implicants Must cover all 1s of the function on K-map Each 1 can be used multiple times in generating the terms of the expression When you group 1’s on a K-map, generating a term, that term is an implicant of the function Prime Implicants Essential Prime Implicants 9/15/09 - L8 Map Manupulation Copyright 2009 - Joanne DeGroat, ECE, OSU3

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Prime Implicant If removal of an any literal from an implicant P results in a product term that is not an implicant of the function, then P is a prime implicant. Let’s take a look at this. 9/15/09 - L8 Map Manupulation Copyright 2009 - Joanne DeGroat, ECE, OSU4

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Implicants Implicants with 2 1s with 4 1s 9/15/09 - L8 Map Manupulation Copyright 2009 - Joanne DeGroat, ECE, OSU5

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Which are prime implicants? Consider A’B’D Remove A’ B’D THUS meets previous statement However, on an n-variable map, the set of prime implicants corresponds to the set of rectangles made up of 2 m squares containing 1s with each rectangle containing as many squares as possible. Thus, A’B’D is not a prime implicant. It is an implicant, just not a prime implicant. 9/15/09 - L8 Map Manupulation Copyright 2009 - Joanne DeGroat, ECE, OSU6

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Prime implicants of the function Have A’D A’B and BD Removal of any literal from any of these terms results in an implicant that is not implicant of the function. They also cover all 1s of the function and are not contained in some larger implicant These are the prime implicants of the function 9/15/09 - L8 Map Manupulation Copyright 2009 - Joanne DeGroat, ECE, OSU7

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Some general statements on P I A single 1 on a map is a prime implicant if is not adjacent to any other 1 of the function. Two adjacent 1s on a map represent a prime implicant, provided that they are not within a rectangle of 4 or more squares containing 1s. Four 1’s that are an implicant are a prime implicant if they are not within a group of 8. 9/15/09 - L8 Map Manupulation Copyright 2009 - Joanne DeGroat, ECE, OSU8

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Essential Prime Implicant A prime implicant that contains a 1 that is not covered by any other prime implicant of the function is an essential prime implicant. IT MUST BE INCLUDED IN ANY MINIMAL REPRESENTATION OF THE FUNCTION. 9/15/09 - L8 Map Manupulation Copyright 2009 - Joanne DeGroat, ECE, OSU9

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Example Consider the example of Fig 2-13 of text. Prime Implicants A’D BD’ A’B Essential Prime Implicants A’D BD’ So F=A’D + BD’ 9/15/09 - L8 Map Manupulation Copyright 2009 - Joanne DeGroat, ECE, OSU10

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Using non-essential prime impicants Consider the example from 2-14 Having Prime Implicants 9/15/09 - L8 Map Manupulation Copyright 2009 - Joanne DeGroat, ECE, OSU11

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continue Include those that are essential Leaves just a single 1 uncovered Have the 2 choices to use for covering it, both of equal size (i.e. same # of literals) Choose either Choose as shown getting F = A’B’C’D’ +BC’D+ABC’ + AB’C + ABD 9/15/09 - L8 Map Manupulation Copyright 2009 - Joanne DeGroat, ECE, OSU12

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Selecting non-essential Selection Rule: Minimize the overlap among prime implicants as much as possible. Make sure each prime implicant selected includes at least one minterm not included in any other prime implicant selected. 9/15/09 - L8 Map Manupulation Copyright 2009 - Joanne DeGroat, ECE, OSU13

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Another problem Problem: Cover largest group of 1s 9/15/09 - L8 Map Manupulation Copyright 2009 - Joanne DeGroat, ECE, OSU14

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Select 1s not covered First Second 9/15/09 - L8 Map Manupulation Copyright 2009 - Joanne DeGroat, ECE, OSU15

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Just 1 more to go have a choice Both are equal Choose 1 Get F = A’C’+ABD+AB’C + A’B’D’ Equal in cost to F = A’C’+ABD+AB’C + B’CD’ 9/15/09 - L8 Map Manupulation Copyright 2009 - Joanne DeGroat, ECE, OSU16

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Don’t cares In many instances in digital systems there are combinations of inputs that you don’t care what the output should be. Example: BCD – You know that the input can only be one of the BDC digit representations and don’t care how the circuit responds for A through F (10 through 15). Results in an incompletely specified function. In examples so far, all the blank squares were actually 0s. 9/15/09 - L8 Map Manupulation Copyright 2009 - Joanne DeGroat, ECE, OSU17

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Don’t care example Consider Very similar but Now have Xs The two prime implicants indicated must be part of the expression Can use the Xs as 0s or 1s as needed 9/15/09 - L8 Map Manupulation Copyright 2009 - Joanne DeGroat, ECE, OSU18

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Using the don’t cares Gives Result F = A’C’+ABD+AC+CD’ 9/15/09 - L8 Map Manupulation Copyright 2009 - Joanne DeGroat, ECE, OSU19

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Product-of-Sums A simple method Simplify F(A,B,C,D)=∑m(0,1,2,5,8,9,10) Map to K-map Include 0s 9/15/09 - L8 Map Manupulation Copyright 2009 - Joanne DeGroat, ECE, OSU20

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Group the 0’s Use groupings to generate F’ F’=AB+CD+BD’ Use DeMorgans (twice) F=(A’+B’)(C’+D’)(B’+D) Which is a product-of-sums representation 9/15/09 - L8 Map Manupulation Copyright 2009 - Joanne DeGroat, ECE, OSU21

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Class 9 summary & assignment Covered section 2-5 Problems for hand in 2-20 2-23 Problems for practice 2-19, 22, 24, 26 Reading for next class: section 2-8 thru 2.11 9/15/09 - L8 Map Manupulation Copyright 2009 - Joanne DeGroat, ECE, OSU22

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