Website: https://vikasdhiman.info/ECE275-Sequential-Logic/

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

- Homework 3 is due on Sept 27th, Monday before class.

- For 2-5 variables use K-Map
- For upto 15 variables, use Quine McCluskey method (NP complete). Complexity grows exponentially with number of input variables
- EXPRESSO-Exact, EXPRESSO-II, CAPPUCCINO are a few of the popular methods that are taught in advanced version of this course.

\(\bar{x}_1\) | \(x_1 \) | ||||
---|---|---|---|---|---|

\(\bar{x}_2\) | \(x_2 \) | \(\bar{x}_2\) | |||

\(\bar{x}_3\) | \(\bar{x}_4\) | 0 | 0 | 0 | 0 |

\(x_4\) | 0 | 1 | 1 | 0 | |

\(x_3\) | 0 | 1 | 1 | 0 | |

\(\bar{x}_4\) | 1 | 1 | 0 | 0 |

\(\bar{x}_1\) | \(x_1 \) | ||||
---|---|---|---|---|---|

\(\bar{x}_2\) | \(x_2 \) | \(\bar{x}_2\) | |||

\(\bar{x}_3\) | \(\bar{x}_4\) | 0 | 0 | 0 | 0 |

\(x_4\) | 0 | 1 | 0 | 0 | |

\(x_3\) | 0 | 1 | 0 | 0 | |

\(\bar{x}_4\) | 1 | 1 | 0 | 0 |

\(\bar{x}_1\) | \(x_1 \) | ||||
---|---|---|---|---|---|

\(\bar{x}_2\) | \(x_2 \) | \(\bar{x}_2\) | |||

\(\bar{x}_3\) | \(\bar{x}_4\) | 0 | 0 | 0 | 0 |

\(x_4\) | 0 | 0 | 1 | 1 | |

\(x_3\) | 0 | 0 | 1 | 0 | |

\(\bar{x}_4\) | 0 | 0 | 0 | 0 |

\( f_1 = x_2 x_4 + g_1 \), Cost = 1 + 1 + 2 + 2 = 6

\( f_2 = \bx_1 x_2 x_4 + g_1 \), Cost = 1 + 1 + 3 + 2= 7

\( f_3 = x_1 x_2 x_4 + x_1 \bx_3 x_4 \), Cost = 2 + 1 + 3 + 3 + 2 = 11

Total Cost: 28

\( g_2 = x_1 x_2 x_4 \), Cost = 1 + 3 = 4

\( g_3 = \bx_1 x_2 x_4 \), Cost = 1 + 3 = 4

\( f_1 = g_1 + g_2 + g_3 \), Cost = 1 + 3 = 4

\( f_2 = g_3 + g_1 \), Cost = 1 + 2 = 3

\( f_3 = g_2 + x_1 \bx_3 x_4 \), Cost = 1 + 1 + 3 + 2 = 7

Total Cost: 26

\(\bar{x}_1\) | \(x_1 \) | ||||
---|---|---|---|---|---|

\(\bar{x}_2\) | \(x_2 \) | \(\bar{x}_2\) | |||

\(\bar{x}_3\) | \(\bar{x}_4\) | 0 | 0 | 1 | 0 |

\(x_4\) | 0 | 0 | 1 | 0 | |

\(x_3\) | 0 | 0 | 1 | 1 | |

\(\bar{x}_4\) | 0 | 0 | 1 | 0 |

\(\bar{x}_1\) | \(x_1 \) | ||||
---|---|---|---|---|---|

\(\bar{x}_2\) | \(x_2 \) | \(\bar{x}_2\) | |||

\(\bar{x}_3\) | \(\bar{x}_4\) | 0 | 0 | 1 | 0 |

\(x_4\) | 0 | 0 | 1 | 0 | |

\(x_3\) | 1 | 1 | 1 | 1 | |

\(\bar{x}_4\) | 0 | 0 | 0 | 0 |

\(\bar{x}_1\) | \(x_1 \) | ||||
---|---|---|---|---|---|

\(\bar{x}_2\) | \(x_2 \) | \(\bar{x}_2\) | |||

\(\bar{x}_3\) | \(\bar{x}_4\) | 0 | 0 | 1 | 0 |

\(x_4\) | 1 | 1 | 1 | 0 | |

\(x_3\) | 0 | 0 | 1 | 0 | |

\(\bar{x}_4\) | 0 | 0 | 1 | 0 |

\(\bar{x}_1\) | \(x_1 \) | ||||
---|---|---|---|---|---|

\(\bar{x}_2\) | \(x_2 \) | \(\bar{x}_2\) | |||

\(\bar{x}_3\) | \(\bar{x}_4\) | 0 | 0 | 1 | 0 |

\(x_4\) | 0 | 0 | 1 | 0 | |

\(x_3\) | 0 | 0 | 1 | 1 | |

\(\bar{x}_4\) | 0 | 0 | 1 | 0 |

\(\bar{x}_1\) | \(x_1 \) | ||||
---|---|---|---|---|---|

\(\bar{x}_2\) | \(x_2 \) | \(\bar{x}_2\) | |||

\(\bar{x}_3\) | \(\bar{x}_4\) | 0 | 0 | 1 | 0 |

\(x_4\) | 0 | 0 | 1 | 0 | |

\(x_3\) | 1 | 1 | 1 | 1 | |

\(\bar{x}_4\) | 0 | 0 | 0 | 0 |

\(\bar{x}_1\) | \(x_1 \) | ||||
---|---|---|---|---|---|

\(\bar{x}_2\) | \(x_2 \) | \(\bar{x}_2\) | |||

\(\bar{x}_3\) | \(\bar{x}_4\) | 0 | 0 | 1 | 0 |

\(x_4\) | 0 | 0 | 1 | 0 | |

\(x_3\) | 1 | 1 | 1 | 0 | |

\(\bar{x}_4\) | 0 | 0 | 1 | 0 |

\[ F_1 = AB + ACD \]
\[ F_2 = AB\bar{C} + \bar{A}CD + ACD \]
\[ F_3 = AB + \bar{A}CD \]

For multiple output functions, we consider only those 1's for EPI that are not present in other fuction maps.

\(\bar{x}_1\) | \(x_1 \) | ||||
---|---|---|---|---|---|

\(\bar{x}_2\) | \(x_2 \) | \(\bar{x}_2\) | |||

\(\bar{x}_3\) | \(\bar{x}_4\) | 0 | 0 | 0 | 0 |

\(x_4\) | 1 | 1 | 1 | 1 | |

\(x_3\) | 0 | 0 | 1 | 0 | |

\(\bar{x}_4\) | 0 | 0 | 0 | 0 |

\(\bar{x}_1\) | \(x_1 \) | ||||
---|---|---|---|---|---|

\(\bar{x}_2\) | \(x_2 \) | \(\bar{x}_2\) | |||

\(\bar{x}_3\) | \(\bar{x}_4\) | 0 | 1 | 1 | 0 |

\(x_4\) | 0 | 0 | 0 | 0 | |

\(x_3\) | 0 | 0 | 1 | 0 | |

\(\bar{x}_4\) | 0 | 1 | 1 | 0 |

- Quine McCluskey can be extended to Multi-output functions: http://web.mit.edu/6.111/www/f2017/handouts/qm.pdf

To avoid static hazards, "close all the gaps" in the K-map by adding extra product terms (or sum terms for POS).

- SOP can only have static-1 hazards.
- POS can only have static-0 hazards.
- More than 2-level circuits needed for dynamic hazards.