ECE 275: More about K-maps

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

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 \( f = 1 \)
Example: \( f = \sum m(0, 1, 2, 3, 7) = \bx_1 + x_1 x_2 x_3\)

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
Find the cost of the \( g = \overline{(x_1\bx_2 + x_3)}(\bx_4 + x_4)\)

Cost = 2 AND + 2 OR + 1 NOT + 9 inputs = 14

Find the cost of the \( f = (x_2 + x_3)(x_3 + x_4)(\bx_1 + \bx_2 + \bx_3 + \bx_4)\)

Cost = 3 OR + 1 AND + (8+3) inputs = 15

K-map neighborhood property

\begin{align} f &= x_1 \bx_3 \bx_4 + x_2 \bx_3 \bx_4 + x_1 \bx_2 \bx_3 & \end{align}
\(\bar{x}_1\) \(x_1 \)
\(\bar{x}_2\) \(x_2 \) \(\bar{x}_2\)
\(\bar{x}_3\)\(\bar{x}_4\) 0 1 1 1
\(x_4\) 0 0 0 1
\(x_3\) 0 0 0 0
\(\bar{x}_4\) 0 0 0 0

Adjacent cells of K-maps differ only by one variable

Karnaugh Maps for 5-variable

Karnaugh Maps for 6-variable

K-Maps for POS minimization

\[ f(x_1, x_2, x_3) = \prod M(4, 5, 6) \]
\[ f(x_1, x_2, x_3) = \prod M(4, 5, 6) \] \[ f(x_1, x_2, x_3) = (\bx_1 + x_3)(\bx_1 + x_2) \]

Cost = 9 = 6 inputs + 2 OR gate + 1 AND gate

\[ f(x_1, x_2, x_3) = \prod M(4, 5, 6) \] \[ \bar{f}(x_1, x_2, x_3) = \sum m(4, 5, 6) \]
\(\bar{x}_1\) \(x_1 \)
\(\bar{x}_2\) \(x_2 \) \(\bar{x}_2\)
\(\bar{x}_3\) 0 0 1
1
\(x_3\) 0 0 0 1
\[ \bar{f} = \color{red}{x_1 \bx_3} + \color{blue}{x_1 \bx_2 } \] \[ f = \color{red}{(\bx_1 + x_3)}\color{blue}{(\bx_1 + x_2)} \]

Minimization process

  1. Draw K-map for \( f \) and \( \bar{f} \)
  2. Find cost for each. Choose the minimum cost implementation.

Incompletely specified functions

\[f(x_1, \dots, x_4) = \sum m(2, 4, 5, 6, 10) + D(12, 13, 14, 15).\]
\[f(x_1, \dots, x_4) = \sum m(2, 4, 5, 6, 10) + D(12, 13, 14, 15).\]
\(\bar{x}_1\) \(x_1 \)
\(\bar{x}_2\) \(x_2 \) \(\bar{x}_2\)
\(\bar{x}_3\)\(\bar{x}_4\) 0 1 d 0
\(x_4\) 0 1 d 0
\(x_3\) 0 0 d 0
\(\bar{x}_4\) 1 1 d 1
\[f = \color{blue}{x_2 \bx_3} + \color{red}{x_3 \bx_4} \]

Cost = 6 inputs + 2 AND + 1 OR = 7

\(\bar{x}_1\) \(x_1 \)
\(\bar{x}_2\) \(x_2 \) \(\bar{x}_2\)
\(\bar{x}_3\)\(\bar{x}_4\) 0 1 d 0
\(x_4\) 0 1 d 0
\(x_3\) 0 0 d 0
\(\bar{x}_4\) 1 1 d 1
\[\bar{f} = \color{blue}{\bx_2 \bx_3} + \color{red}{\bx_3 x_4} \] \[f = \color{blue}{(x_2 + x_3)} \color{red}{(x_3 + \bx_4)} \]

Cost = 6 inputs + 2 OR + 1 AND = 7

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