05. Minimum SOP and POS

Problem Statement

A single-output digital system with four inputs (a,b,c,d) generates a logic-1 when 2, 7, or 15 appears on the inputs, and a logic-0 when 0, 1, 4, 5, 6, 9, 10, 13, or 14 appears. The input conditions for the numbers 3, 8, 11, and 12 never occur in this system. For example, 7 corresponds to a,b,c,d being set to 0,1,1,1, respectively.

Determine the output out_sop in minimum SOP form, and the output out_pos in minimum POS form.

Solution

Truth Table

a	b	c	d	out
0	0	0	0	0
0	0	0	1	0
0	0	1	0	1
0	0	1	1	X
0	1	0	0	0
0	1	0	1	0
0	1	1	0	0
0	1	1	1	1
1	0	0	0	X
1	0	0	1	0
1	0	1	0	0
1	0	1	1	X
1	1	0	0	X
1	1	0	1	0
1	1	1	0	0
1	1	1	1	1

SOP

OUT = $\Sigma _m (2, 7, 8) + d_m(3,8,12)$

$= \overline{a}\overline{b}c+cd$

POS

OUT = $\Pi _M (0, 1, 4, 5, 6, 9, 10, 13, 14) . D_m(3,8,12)$

$= c(\overline{a}+b)(\overline{b}+d)$

Find the verilog file here