ECE 275: Quine McCluskey method

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\[ \newcommand{\bx}{\bar{x}} \newcommand{\by}{\bar{y}} \newcommand{\bz}{\bar{z}} \]

K-map 5-var example

\[ f (x_1 , \dots , x_5 ) = \sum m(1, 4, 6, 7, 9, 10, 12, 15, 17, 19, \\ 20, 23, 25, 26, 27, 28, 30, 31) + D(8, 16, 21, 22). \]

Karnaugh Maps terminology

  • Literal : a single variable or its complement.
  • Implicant: A product term that indicates for which \( f = 1 \)
    All minterms are implicants.
  • Prime Implicant (PI): If an implicant cannot be "combined" into fewer literals.
  • Essential Prime Implicant (EPI): A PI that is the only PI to cover some 1 on the K-map.
  • Cover : PI's that account for all \( f = 1 \).
  • Cost : Number of gates and inputs excluding input complements.
Example: \( f = \sum m(0, 1, 2, 3, 7) = \bx_1 + x_1 x_2 x_3\)
\(\bar{x_1}\) \(x_1 \)
\(\bar{x_2}\) \(x_2 \) \(\bar{y}\)
\(\bar{x_3}\) 1 1 0 0
\(x_3\) 1 1+1 1 0
  • Literals are \( \bx_1, x_1, x_2, x_3\)
  • Implicants are \( \bx_1 \), \(\bx_1 x_2 x_3, \dots \), \( x_1 x_2 x_3\)
  • PI's are \( \bx_1 \) and \( x_2 x_3 \)
  • EPI's are \( \bx_1 \) and \( x_2 x_3 \)
  • Cost is 9 = 6 inputs + 2 AND gate + 1 OR gate

Quine McCluskey method

\[ f (x 1 , \dots , x 4 ) = \sum m(0, 4, 8, 10, 11, 12, 13, 15) \]

Finding PIs

\[ P = \{ 11*0, 101*, 110*, 1*11, 11*1, **00 \} \]

Minimum Cover

\[ P = \{ 11*0, 101*, 110*, 1*11, 11*1, **00 \} \] \[ f = x_1 \bx_2 x_3 + x_1 x_2 x_4 + \bx_3 \bx_4 \]

Quine McCluskey method

\[ f (x 1 , \dots , x 4 ) = \sum m(0, 2, 5, 6, 7, 8, 9, 13) + D(1, 12, 15) \]

Finding PIs

\[ P = \{ 00*0, 0*10, 011*, *00*, **01, 1*0*, *1*1\} \]

Minimum Cover

\[ P = \{ 00*0, 0*10, 011*, *00*, **01, 1*0*, *1*1\} \] \[ f = \bx_1 x_3 \bx_4 + \bx_2 \bx_3 + x_2 x_4\]

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