## 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?

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