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Simplification of switching functions

Simplification of switching functions. Simplify – why? Switching functions map to switching circuits Simpler function  simpler circuit Reduce hardware complexity Reduce size and increase speed by reducing number of gates Simplify – how? Using the postulates Ad-hoc.

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Simplification of switching functions

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  1. Simplification of switching functions • Simplify – why? • Switching functions map to switching circuits • Simpler function  simpler circuit • Reduce hardware complexity • Reduce size and increase speed by reducing number of gates • Simplify – how? • Using the postulates • Ad-hoc

  2. Simplification of switching functions • Simplify – what? • SOP/POS form has products/sums and literals • Literal: each appearance of a variable or its complement • Minimize number of sums/products • Reduces total gate count • Minimize number of variables in each sum/product • Reduces number of inputs to each gate • PLDs have fixed # of inputs; only the number of terms need to be minimized there

  3. Simplification of switching functions

  4. Simplification using postulates

  5. Simplification using Karnaugh maps

  6. Karnaugh maps • Karnaugh map (also K-map) is a graphic tool, pictorial representation of truth table • Extension of the concepts of truth table, Venn diagram, minterm • Transition from Venn diagram to minterm

  7. Karnaugh maps • Adjacencies are preserved when going from c) to d) • They are the same, only the areas are made equal in d), which preserves adjacencies • Subscripts are dropped in e); realize that 2&3 is A; 1&3 is B • In f) the labels change and become 0 and 1 • Each square of the K-map is 1 row of the TT

  8. Karnaugh maps • Might start with rectangles initially and get the same result  A B  • Each square of the K-map is 1 row of the TT

  9. Karnaugh maps • One to one correspondence between K-map squares and maxterms A A+B  M0 = m0 = AB B A A+B  M3 = m3 = AB B

  10. Karnaugh maps • One to one correspondence between K-map squares and maxterms A A+B  M2 = m2 = AB B A A+B  M1 = m1 = AB B

  11. 3-variable K-maps

  12. 3-variable K-maps • Constructing 3-variable K-maps A A B 0 1 1 0 B 0 flip  0 1 1 C = 0 C = 1 abutt CA B 00 01 11 10 0 1

  13. 3-variable K-maps • Constructing 3-variable K-maps A A B 0 1 CB 1 0 0 C = 0 00 1 01 C = 0 11 A 10 B 0 1 1 C = 1 0

  14. 4-variable K-maps

  15. 5-variable K-maps

  16. 5-variable K-maps

  17. 6-variable K-maps

  18. 6-variable K-maps

  19. Plotting functions in canonical form

  20. Plotting functions in canonical form

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