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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
Copyright 2009 - Joanne DeGroat, ECE, OSU 2
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
9/15/09 - L8 Map Manupulation
Copyright 2009 - Joanne DeGroat, ECE, OSU 3
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, OSU 4
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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Copyright 2009 - Joanne DeGroat, ECE, OSU 8
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, OSU 9
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, OSU 10
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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