Announcments

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

Limitations of this course

• 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.

Multi output functions: Ex 1

$$f_1$$
$$\bar{x}_1$$ $$x_1$$
$$\bar{x}_2$$ $$x_2$$ $$\bar{x}_2$$
$$\bar{x}_3$$$$\bar{x}_4$$ 0 0 0 0
$$x_4$$ 0 1 1 0
$$x_3$$ 0 1 1 0
$$\bar{x}_4$$ 1 1 0 0
$$f_2$$
$$\bar{x}_1$$ $$x_1$$
$$\bar{x}_2$$ $$x_2$$ $$\bar{x}_2$$
$$\bar{x}_3$$$$\bar{x}_4$$ 0 0 0 0
$$x_4$$ 0 1 0 0
$$x_3$$ 0 1 0 0
$$\bar{x}_4$$ 1 1 0 0
$$f_3$$
$$\bar{x}_1$$ $$x_1$$
$$\bar{x}_2$$ $$x_2$$ $$\bar{x}_2$$
$$\bar{x}_3$$$$\bar{x}_4$$ 0 0 0 0
$$x_4$$ 0 0 1 1
$$x_3$$ 0 0 1 0
$$\bar{x}_4$$ 0 0 0 0

Ex 1: Soln

Separately optimized
$$g_1 = \bx_1 x_3 \bx_4$$, Cost = 1 + 3 = 4
$$f_1 = x_2 x_4 + g_1$$, 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

Jointly optimized
$$g_1 = \bx_1 x_3 \bx_4$$, Cost = 1 + 3 = 4
$$g_2 = x_1 x_2 x_4$$, 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

Multi output functions: Ex 2

$$f_1$$
$$\bar{x}_1$$ $$x_1$$
$$\bar{x}_2$$ $$x_2$$ $$\bar{x}_2$$
$$\bar{x}_3$$$$\bar{x}_4$$ 0 0 1 0
$$x_4$$ 0 0 1 0
$$x_3$$ 0 0 1 1
$$\bar{x}_4$$ 0 0 1 0
$$f_2$$
$$\bar{x}_1$$ $$x_1$$
$$\bar{x}_2$$ $$x_2$$ $$\bar{x}_2$$
$$\bar{x}_3$$$$\bar{x}_4$$ 0 0 1 0
$$x_4$$ 0 0 1 0
$$x_3$$ 1 1 1 1
$$\bar{x}_4$$ 0 0 0 0
$$f_3$$
$$\bar{x}_1$$ $$x_1$$
$$\bar{x}_2$$ $$x_2$$ $$\bar{x}_2$$
$$\bar{x}_3$$$$\bar{x}_4$$ 0 0 1 0
$$x_4$$ 1 1 1 0
$$x_3$$ 0 0 1 0
$$\bar{x}_4$$ 0 0 1 0

Ex 3

$$f_1$$
$$\bar{x}_1$$ $$x_1$$
$$\bar{x}_2$$ $$x_2$$ $$\bar{x}_2$$
$$\bar{x}_3$$$$\bar{x}_4$$ 0 0 1 0
$$x_4$$ 0 0 1 0
$$x_3$$ 0 0 1 1
$$\bar{x}_4$$ 0 0 1 0
$$f_2$$
$$\bar{x}_1$$ $$x_1$$
$$\bar{x}_2$$ $$x_2$$ $$\bar{x}_2$$
$$\bar{x}_3$$$$\bar{x}_4$$ 0 0 1 0
$$x_4$$ 0 0 1 0
$$x_3$$ 1 1 1 1
$$\bar{x}_4$$ 0 0 0 0
$$f_3$$
$$\bar{x}_1$$ $$x_1$$
$$\bar{x}_2$$ $$x_2$$ $$\bar{x}_2$$
$$\bar{x}_3$$$$\bar{x}_4$$ 0 0 1 0
$$x_4$$ 0 0 1 0
$$x_3$$ 1 1 1 0
$$\bar{x}_4$$ 0 0 1 0

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

EPIs for Multi-output functions

For multiple output functions, we consider only those 1's for EPI that are not present in other fuction maps.
$$f_1$$
$$\bar{x}_1$$ $$x_1$$
$$\bar{x}_2$$ $$x_2$$ $$\bar{x}_2$$
$$\bar{x}_3$$$$\bar{x}_4$$ 0 0 0 0
$$x_4$$ 1 1 1 1
$$x_3$$ 0 0 1 0
$$\bar{x}_4$$ 0 0 0 0
$$f_2$$
$$\bar{x}_1$$ $$x_1$$
$$\bar{x}_2$$ $$x_2$$ $$\bar{x}_2$$
$$\bar{x}_3$$$$\bar{x}_4$$ 0 1 1 0
$$x_4$$ 0 0 0 0
$$x_3$$ 0 0 1 0
$$\bar{x}_4$$ 0 1 1 0

Hazards: Example fixed

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.

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