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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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Presentation on theme: "9/15/09 - L8 Map ManupulationCopyright 2009 - Joanne DeGroat, ECE, OSU1 Map Manupulation Optimization, terms and don’t cares."— Presentation transcript:

1 9/15/09 - L8 Map ManupulationCopyright Joanne DeGroat, ECE, OSU1 Map Manupulation Optimization, terms and don’t cares

2 9/15/09 - L8 Map Manupulation Copyright 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

3 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 Joanne DeGroat, ECE, OSU3

4 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 Joanne DeGroat, ECE, OSU4

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

6 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 Joanne DeGroat, ECE, OSU6

7 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 Joanne DeGroat, ECE, OSU7

8 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 Joanne DeGroat, ECE, OSU8

9 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 Joanne DeGroat, ECE, OSU9

10 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 Joanne DeGroat, ECE, OSU10

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

12 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 Joanne DeGroat, ECE, OSU12

13 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 Joanne DeGroat, ECE, OSU13

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

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

16 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 Joanne DeGroat, ECE, OSU16

17 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 Joanne DeGroat, ECE, OSU17

18 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 Joanne DeGroat, ECE, OSU18

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

20 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 Joanne DeGroat, ECE, OSU20

21 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 Joanne DeGroat, ECE, OSU21

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


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