ECE 275: Multi-level synthesis

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

\[ \newcommand{\bx}{\bar{x}} \newcommand{\by}{\bar{y}} \newcommand{\bz}{\bar{z}} \newcommand{\bA}{\bar{A}} \newcommand{\bB}{\bar{B}} \newcommand{\bC}{\bar{C}} \newcommand{\bD}{\bar{D}} \newcommand{\bE}{\bar{E}} \]

Announcments

  • Homework 4 has only four problems and is due on Oct 1st Friday by midnight.
  • We will review material covered so far on Oct 4th, please email me topics that you want to be reviewed.
  • Midterm is on Oct 6th, 9 AM in class. All problems will be variations of the homework problems.

Gates review

AND gate
\( f(x_1, x_2) = x_1 \cdot x_2 \)
OR gate
\( f_{\text{OR}}(x_1, x_2) = x_1 + x_2 \)
NOT gate
\( L_{\text{NOT}}(x_1) = \bar{x}_1 \)
NAND gate
\( Q = \overline{x_1 \cdot x_2} \)
NOR gate
\( Q = \overline{x_1 + x_2} \)
XOR gate
\( Q = \bx_1 x_2 + x_1 \bx_2 = x_1 \oplus x_2 \)

Approaches for Multi-level synthesis

  • Factorization
  • Functional decomposition

Functional decomposition Ex1

\[ f = \bx_1 x_2 x_3 + x_1 \bx_2 x_3 + \bx_1 \bx_2 x_4 + x_1 x_2 x_4 \]

Functional decomposition Ex1

\[ f = \bx_1 x_2 x_3 + x_1 \bx_2 x_3 + \bx_1 \bx_2 x_4 + x_1 x_2 x_4 \]
\(\bar{x}_1\) \(x_1 \)
\(\bar{x}_2\) \(x_2 \) \(\bar{x}_2\)
\(\bar{x}_3\)\(\bar{x}_4\) 0 0 0 0
\(x_4\) 1 0 1 0
\(x_3\) 1 1 1 1
\(\bar{x}_4\) 0 1 0 1

Functional decomposition Ex2

\[ f = \sum m(0, 6, 8, 14) \]
\(\bar{x}_1\) \(x_1 \)
\(\bar{x}_2\) \(x_2 \) \(\bar{x}_2\)
\(\bar{x}_3\)\(\bar{x}_4\) 1 0 0 1
\(x_4\) 0 0 0 0
\(x_3\) 0 0 0 0
\(\bar{x}_4\) 0 1 1 0

Functional decomposition Ex3

\(\bx_5 \)
\(\bar{x}_1\) \(x_1 \)
\(\bar{x}_2\) \(x_2 \) \(\bar{x}_2\)
\(\bar{x}_3\)\(\bar{x}_4\) 1 0 0 0
\(x_4\) 0 1 1 1
\(x_3\) 1 0 0 0
\(\bar{x}_4\) 0 1 1 1
\(x_5 \)
\(\bar{x}_1\) \(x_1 \)
\(\bar{x}_2\) \(x_2 \) \(\bar{x}_2\)
\(\bar{x}_3\)\(\bar{x}_4\) 0 0 0 0
\(x_4\) 1 1 1 1
\(x_3\) 0 0 0 0
\(\bar{x}_4\) 1 1 1 1

Functional decomposition Ex3

\(\bx_5 \)
\(\bar{x}_1\) \(x_1 \)
\(\bar{x}_2\) \(x_2 \) \(\bar{x}_2\)
\(\bar{x}_3\)\(\bar{x}_4\) 1 0 0 0
\(x_4\) 0 1 1 1
\(x_3\) 1 0 0 0
\(\bar{x}_4\) 0 1 1 1
\(x_5 \)
\(\bar{x}_1\) \(x_1 \)
\(\bar{x}_2\) \(x_2 \) \(\bar{x}_2\)
\(\bar{x}_3\)\(\bar{x}_4\) 0 0 0 0
\(x_4\) 1 1 1 1
\(x_3\) 0 0 0 0
\(\bar{x}_4\) 1 1 1 1

Functional decomposition Ex4

\(\bA \)
\(\bB\) \(B \)
\(\bC\) \(C \) \(\bC\)
\(\bD\)\(\bE\) 0 1 1 1
\(E\) 1 0 0 0
\(D\) 0 1 1 1
\(\bE\) 1 0 0 0
\(\bA \)
\(\bB\) \(B \)
\(\bC\) \(C \) \(\bC\)
\(\bD\)\(\bE\) 1 0 0 0
\(E\) 0 1 0 0
\(D\) 1 0 0 0
\(\bE\) 0 1 0 0

Thanks, Questions, Feedback?

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