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Karnaugh Map

Karnaugh Map

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Karnaugh Map

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  1. Karnaugh Map • Introduction • Venn Diagram • 2 variable K-map • 3 variable K-map • 4 variable K-map • K-map simplification MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

  2. Karnaugh Map • Convert to Minterm Form • Simplest SOP expression • Produce POS expression • Don’t care condition MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

  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 MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

  4. Karnaugh Map – Venn Diagram • Venn Diagram represent minterm space • Example: two variables (4 minterm) MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

  5. Karnaugh Map – Venn Diagram • Each minterm set represent Boolean function Example: MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

  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 MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

  7. Karnaugh Map – 2 variable • Map form that can be drawn for 2 variable (a,b) MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

  8. Karnaugh Map – 2 variable • Map form that can be drawn for 2 variable (a,b) – cont.. MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

  9. Karnaugh Map – 2 variable • Equivalent labeling method MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

  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 MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

  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 MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

  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. MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

  13. Karnaugh Map – 4 variable • There are 16 cell in 4 variable K-map (w,x,y,z) MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

  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 MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

  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 MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

  16. Karnaugh Map – 5 variable • It is arrange similar to 4 variable K-map • Similar square on each adjacent map MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

  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, MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

  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 MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

  19. Simplification using Karnaugh Map • Example: MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

  20. Simplification using Karnaugh Map • Get the ‘product’ for each cluster for adjacent minterm (cluster with power of two size) for given function. • Example: MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

  21. Simplification using Karnaugh Map • There are two minterm cluster A and B where (for previous example) MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

  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 MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

  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 MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

  24. Simplification using Karnaugh Map • Other types of cluster in 4 variable K-map case are: MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

  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 MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

  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 MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

  27. Conversion to Minterm Form • Example: MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

  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 MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR