9/15/09 - l8 map manupulationcopyright 2009 - joanne degroat, ece, osu1 map manupulation...

9/15/09 - L8 Map Manupulation Copyright 2009 - Joanne DeGroat, ECE, OSU 1

Map ManupulationOptimization, terms and don’t cares

9/15/09 - L8 Map Manupulation

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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

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

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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.

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Implicants Implicants with 2 1s with 4 1s

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A’B’DA’BDABDA’BC’A’BCA’BD’A’C’DA’CDBC’DBCD

A’DA’BBD

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 2m 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.

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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

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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.

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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.

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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

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Using non-essential prime impicants Consider the example

from 2-14

Having Prime Implicants

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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

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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.

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Another problem Problem:

Cover largest group of 1s

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Select 1s not covered First

Second

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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’

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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.

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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

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Using the don’t cares Gives

Result F = A’C’+ABD+AC+CD’

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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

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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

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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

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