Plotting functions not in canonical form

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Plotting functions not in canonical form

• Plot the function f(a, b, c) a bc
• ab a ab
• c 00 01 11 10 c 00 01 11
  10
• 0 1 1 0 0
  2 6 4
• 1 1 1 1 1 1
  3 7 5
• b
• The squares are numbered derive the canonical
  form

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5-variable K-maps - alternative
0
1
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6-variable K-maps - alternative
00
01

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Simplifying functions using K-maps

• Why is simplification possible
• Logically adjacent minterms are physically
  adjacent on the K-map
• Adjacent minterms can be combined by eliminating
  the common variable
• abc and abc are adjacent
• abc abc bc ? variable a eliminated
• Done by drawing on the map a ring around the
  terms that can be combined

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Simplifying functions using K-maps
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Simplifying functions using K-maps
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Simplifying functions using K-maps

• Definition of terms
• Implicant ? product term that can be used to
  cover minterms
• Prime implicant ? implicant not covered by any
  other implicant
• Essential prime implicant ? a prime implicant
  that covers at least one minterm not covered by
  any other prime implicant
• Cover ? set of prime implicants that cover each
  minterm of the function
• Minimizing a function ? finding the minimum cover

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Simplifying functions using K-maps

• Definition of terms
• Implicants

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Simplifying functions using K-maps

• Definition of terms
• Prime implicants only B and AC
• Essential prime implicants B and AC
• Cover B, AC

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Simplifying functions using K-maps

• Definition of terms
• Implicate ? sum term that can be used to cover
  maxterms (0s on the K-map)
• Prime implicate ? implicate not covered by any
  other implicate
• Essential prime implicate ? a prime implicate
  that covers at least one maxterm not covered by
  any other prime implicate
• Cover ? set of prime implicates that cover each
  maxterm of the function

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Simplifying functions using K-maps

• Algorithm 1
• Fast and easy, not optimal

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Simplifying functions using K-maps

• Algorithm 2
• More work than the first
• Can give better results, because all prime
  implicants are considered
• Still not optimal

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Simplifying functions using K-maps

• Algorithm 2
• 1 Identify all PIs

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Simplifying functions using K-maps

• Algorithm 2
• 2 Identify EPIs

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Simplifying functions using K-maps

• Algorithm 2
• 3 Select cover

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The Quine-McCluskey minimization method

• Tabular
• Systematic
• Can handle a large number of variables
• Can be used for functions with more than one
  output

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The Q-M minimization method
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The Q-M minimization method
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The Q-M minimization method
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The Q-M minimization method

• Combine minterms from List 1 into pairs in List 2
• Take pairs from adjacent groups only, that differ
  in 1 bit
• Combine entries from List 2 into pairs in List 3

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The Q-M minimization method
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The Q-M minimization method
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The Q-M minimization method
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The Q-M minimization method