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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
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


Simplification of switching functions1
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





Karnaugh maps
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


Karnaugh maps1
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


Karnaugh maps2
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


Karnaugh maps3
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


Karnaugh maps4
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



3 variable k maps1
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


3 variable k maps2
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