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

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

3. Copyright 2009 - Joanne DeGroat, ECE, OSU
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, OSU

4. Copyright 2009 - Joanne DeGroat, ECE, OSU
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, OSU

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

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

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, OSU

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, OSU

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, OSU

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

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, OSU

12. Copyright 2009 - Joanne DeGroat, ECE, OSU
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, OSU

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, OSU

14. Copyright 2009 - Joanne DeGroat, ECE, OSU
Another problem
Problem:
Cover largest group of 1s
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Copyright Joanne DeGroat, ECE, OSU

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

16. Copyright 2009 - Joanne DeGroat, ECE, OSU
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, OSU

17. Copyright 2009 - Joanne DeGroat, ECE, OSU
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, OSU

18. Copyright 2009 - Joanne DeGroat, ECE, OSU
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, OSU

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

20. Copyright 2009 - Joanne DeGroat, ECE, OSU
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, OSU

21. Copyright 2009 - Joanne DeGroat, ECE, OSU
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, OSU

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 2.11
9/15/09 - L8 Map Manupulation
Copyright Joanne DeGroat, ECE, OSU