1. ECE2030 Introduction to Computer Engineering Lecture 7: Simplification using K-map
Prof. Hsien-Hsin Sean Lee
School of Electrical and Computer Engineering
Georgia Tech

2. Hamming Distance The count of bits different in two binary patterns
Examples:
Dh(1001, 0101) = 2
Dh(0xADF4, 0x9FE3) = ??
Unit-Distance Codes
Reduce errors during transmission such as rotary positional sensor
E.g. Gray Code

3. Gray Code Construction
000
001
011
010
110
111
101
100 1 1 00 01 11 10 1 1 1 10 11 01 00 1
100
101
111
110
010
011
001
000

4. Gray Code Binary Encoding Gray Code Encoding Decimal b3 b2 b1 b0 g3 g21 2 3 4 5 6 7 8 9 10 11 12 13 14 15

5. Rotary Position Sensor

6. Karnaugh Map (K-Map)
A graphical map method to simplify Boolean function up to 6 variables
A diagram made up of squares
Each square represents one minterm (or maxterm) of a given Boolean function

7. Karnaugh Map Examples
Note that the Hamming Distance between adjacent columns or adjacent rows
(including cyclic ones) must be 1 for simplification purposes

8. Karnaugh Map
Adjacent columns or rows allow grouping of minterms (maxterms)
for simplification

9. Implicant Definition In K-map, an Implicant is
A product term is an Implicant of a Boolean function if the function has an output 1 for all minterms of the product term.
In K-map, an Implicant is
bubble covers only 1 (bubble size must be a power of 2) CD 00 01 11 10 1 AB

10. Prime Implicant Definition In K-map, a Prime Implicant (PI) is
If the removal of any literal from an implicant I results in a product term that is not an implicant of the Boolean function, then I is an Prime Implicant.
Examples
BCD is an implicant, but CD or BD or BC do not imply a 1 in this function; BCD is a PI
B’C’D is an implicant, but B’C’ is not an implicant, thus B’C’D is not a PI
In K-map, a Prime Implicant (PI) is
bubble that is expanded as big as possible (bubble size must be a power of 2) CD 00 01 11 10 1 AB

11. Essential Prime Implicant
Definition
If a minterm of a Boolean function is included in only one PI, then this PI is an Essential Prime Implicant.
In K-map, an Essential Prime Implicant is
Bubble that contains a 1 covered only by itself and no other PI bubbles CD 00 01 11 10 1 AB

12. Non-Essential Prime Implicant
Definition
A Non-Essential Prime Implicant is a PI that is not an Essential PI.
In K-map, an Non-Essential Prime Implicant is
A 1 covered by more than one PI bubble CD 00 01 11 10 1 AB

13. Simplification for SOP
Form K-Map for the given Boolean function
Identify all Essential Prime Implicants for 1’s in the K-map
Identify non-Essential Prime Implicants in the K-map for the 1’s which are not covered by the Essential Prime Implicants
Form a sum-of-products (SOP) with all Essential Prime Implicants and the necessary non-Essential Prime Implicants to cover all 1’s

14. Example for SOP Identify all the essential PIs for 1’s
Identify the non-essential PIs to cover 1’s
Form an SOP based on the selected PIs BC 00 01 11 10 1 A

15. Example for SOP Identify all the essential PIs for 1’s
Identify the non-essential PIs to cover 1’s
Form an SOP based on the selected PIs CD 00 01 11 10 1 AB

16. Example for SOP Identify all the essential PIs for 1’s
Identify the non-essential PIs to cover 1’s
Form an SOP based on the selected PIs CD 00 01 11 10 1 AB

17. Prime Implicants
All the prior definitions apply to ‘0’ (or maxterm) as well
Consider these implicants imply a ‘0’ output

18. Simplification for POS
Form K-Map for the given Boolean function
Identify all Essential Prime Implicants for 0’s in the K-map
Identify non-Essential Prime Implicants in the K-map for the 0’s which are not covered by the Essential Prime Implicants
Form a product-of-sums (POS) with all Essential Prime Implicants and the necessary non-Essential Prime Implicants to cover all 0’s

19. Example for POS Identify all the essential PIs for 0’s
Identify the non-essential PIs to cover 0’s
Form an POS based on the selected PIs BC 00 01 11 10 1 A

20. Example for POS Identify all the essential PIs for 0’s
Identify the non-essential PIs to cover 0’s
Form an POS based on the selected PIs CD 00 01 11 10 1 AB

21. Don’t Care Condition  X
Don’t care (X)
Those input combinations which are irrelevant to the target function (i.e. If the input combination signals can be guaranteed never occur)
Can be used to simplify Boolean equations, thus simply logic design
In K-map
Use X to express Don’t Care in the map
Don’t care can be bubbled as 1 or 0 depending on SOP or POS simplification to result into bigger bubble

22. Don’t Care Example  BCD to Gray Code
BCD stands for Binary Coded Decimal
Each digit of a decimal number is represented by one code
BCD only encodes from 0 to 9
Require 4 bits
Only use 10 out of 16 encoding space
BCD-coded input
Gray Code Output
Decimal b3 b2 b1 b0 g3 g2 g1 g0 1 2 3 4 5 6 7 8 9 10 X 11 12 13 14 15

23. BCD to Gray Code (Do not use Don’ Care)
BCD stands for Binary Coded Decimal
Each digit of a decimal number is represented by one code
BCD only encodes from 0 to 9
Require 4 bits
Only use 10 out of 16 encoding space
BCD-coded input
Gray Code Output
Decimal b3 b2 b1 b0 g3 g2 g1 g0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

24. What g2 is?
g2 = F(b3, b2, b1, b0)

25. Example: g2 of BCD-to-Gray
Without using Don’t Care
Take Don’t Care into account
b1b0
b1b0 00 01 11 10 1
b3b2 00 01 11 10 1 X
b3b2

26. Another Example of Don’t Care (SOP)CD 00 01 11 10 1 X AB

27. Another Example of Don’t Care (POS)CD 00 01 11 10 1 X AB

28. Use Karnaugh Map in B5 or B6
2 K-maps are to be constructed (2 submaps)
Consider one submap is on top of the other
Cells that occupy the same relative position in 2 maps are considered adjacent
Bubble can be constructed in vertical dimension when stacking up 2 maps
In B6
4 K-maps are to be constructed (4 submaps)
Similar to B5 , yet another dimension needs to be considered

29. Karnaugh Map Example in B5CD CD 00 01 11 10 1 00 01 11 10 1 AB AB
E = 0
E = 1

30. Karnaugh Map Example in B6YZ YZ 00 01 11 10 1 00 01 11 10 1 WX WX
U,V=0,0
U,V=0,1 YZ YZ 00 01 11 10 1 X 00 01 11 10 1 WX WX
U,V=1,0
U,V=1,1

31. Minimal SOP and POS w/ Don’t care
Could lead to different function if the same x is treated as 1 in SOP but as 0 in POS.
Even though the final Boolean functions could be different, but for those ones with non-don’t care square, correct functionality was maintained. 1 x B A
F = Ā (SOP)
F = B (POS)
F = Ā (POS)

32. Use Karnaugh Map in B5 or B6
It is getting hard and complicated
Eye-ball simplification on several submaps is error-prone, leading to sub-optimal results
Do we have a better algorithm?
Quine-McCluskey Method