1. MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR
Karnaugh Map
Introduction
Venn Diagram
2 variable K-map
3 variable K-map
4 variable K-map
K-map simplification
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2. MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR
Karnaugh Map
Convert to Minterm Form
Simplest SOP expression
Produce POS expression
Don’t care condition
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3. Karnaugh Map -Introduction
Systematic method to get simplest Boolean SOP expression
Objective: Minimum number of literal
Dramatic technique based on special Venn Diagram Form
Advantage: Easy with visual aid
Disadvantage: Limited to five or six variables
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4. Karnaugh Map – Venn Diagram
Venn Diagram represent minterm space
Example: two variables (4 minterm)
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5. Karnaugh Map – Venn Diagram
Each minterm set represent Boolean function
Example:
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6. Karnaugh Map – 2 variable
K-map is abstract form, Venn diagram is arranged as square matrix, where
Each square represent one minterm
Adjacent square is always differentiated with one literal (therefore theorem a+a’ can be used)
For two variable case (e.g. variable a, b), map can be drawn as
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7. Karnaugh Map – 2 variable
Map form that can be drawn for 2 variable (a,b)
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8. Karnaugh Map – 2 variable
Map form that can be drawn for 2 variable (a,b) – cont..
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9. Karnaugh Map – 2 variable
Equivalent labeling method
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10. Karnaugh Map – 2 variable
K-map for a function is determined by placing sign
1 on equivalent square with minterm
0 vice versa
For example: Carry & Sum for half adder
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11. Karnaugh Map – 3 variable
There are 8 minterm for three variable (a,b,c). Therefore, 8 cell in three variable K-map
The above array ensure that minterm in adjacent cell only has one literal difference
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12. Karnaugh Map – 3 variable
There are wrap-around in 3 variable K-map
a’b’c’(m0) is as neighbor next to a’bc’(m2)
ab’c’(m4) is as neighbor next to abc’(m6)
Each cell in 3 variable K-map contains three adjacent neighbor. Generally, each cell in K-map n variable contain n adjacent neighbor.
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13. Karnaugh Map – 4 variable
There are 16 cell in 4 variable K-map (w,x,y,z)
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14. Karnaugh Map – 4 variable
There are 2 wrap-around: vertical & horizontal
Each cell has 4 adjacent neighbor. For example: cell m0 is a neighbor of m1, m2, m4 and m8
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15. Karnaugh Map – 5 variable
Map for more than 4 variable are more complicated since the geometry (hypercube configuration) for adjacent neighbor is greater.
For five variable; e.g. vwxyz there is 25=32 squares
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16. Karnaugh Map – 5 variable
It is arrange similar to 4 variable K-map
Similar square on each adjacent map
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17. Simplification using Karnaugh Map
Based on Theorem:
A+A’=1
In K-map, each cell contains ‘1’ representing minterm for the given function
Each adjacent cell cluster contains ‘1’ (the cluster must have a power of two size which is 1,2,4,8….) then get the simplified value for each cluster
grouped 2 adjacent squares will eliminate 1 variable, grouped 4 adjacent squares will eliminate 2 variable, grouped 8 adjacent squares will eliminate 3 variable, grouped 2n adjacent squares will eliminate n variable,
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18. Simplification using Karnaugh Map
Grouped largest possible squares (minterm) in a cluster
The grater the cluster, the lesser the number of literals in your answer
Get the number of small cluster to gather all squares (minterm) for the function
The lesser the cluster, the lesser the number of ‘product’ and the minimize function
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19. Simplification using Karnaugh Map
Example:
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20. Simplification using Karnaugh Map
Get the ‘product’ for each cluster for adjacent minterm (cluster with power of two size) for given function.
Example:
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21. Simplification using Karnaugh Map
There are two minterm cluster A and B where (for previous example)
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22. Simplification using Karnaugh Map
Each ‘product’ for cluster, w’xy’ and wy represent sum-of-minterm in that cluster
Boolean Function is sum-of-product which represent all minterm cluster for that function
F(w,x,y,z)=A+B=w’xy’+wy
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23. Simplification using Karnaugh Map
Greater cluster produce ‘product’ with small number of literal. In 4 variable K-map case:
1 cell = 4 literal, example: wxyz, w’xy’z
2 cell = 3 literal, example: wxy, w’y’z’
4 cell = 2 literal, example: wx, x’y
8 cell = 1 literal, example: w, y’, z’
16 cell = no literal example: 1
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24. Simplification using Karnaugh Map
Other types of cluster in 4 variable K-map case are:
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25. Simplification using Karnaugh Map
Minterm cluster must be
Square, and
Power of two size
If not, it is not a certified cluster. Examples of non certified cluster are
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26. Conversion to Minterm Form
Function is easy to draw in K-map when function is given in SOP or SOM canonical form
How if it is not in sum-of-minterm form?
Convert it to sum-of-product
Elaborate SOP expression to SOM expression, or fill SOP expression directly to K-map
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27. Conversion to Minterm Form
Example:
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28. SOP Expression Simplification
To get the simplest SOP expression from K-map, you will need
Minimum number of literal for each ‘product’, and
Minimum number of ‘product’
This can be achieved in K-map by using
Largest possible number of minterm in one cluster (i.e. prime implicant (PI))
No extra cluster (i.e. essential prime implicant (epi))
Implicant: is a "covering" (sum term or product term) of one or more in a sum of product (or a maxterm in a product of sum) of a boolean function
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