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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

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