Chapter B, Problem 40E

Explanation of Solution

Truth table:

Present state			Next state			Required input
Q1	Q2	Q3	Q1’	Q2’	Q3’	d1	d2	d3
0	0	0	0	0	1	0	0	1
0	0	1	0	1	1	0	1	1
0	1	1	0	1	0	0	1	0
0	1	0	1	1	0	1	1	0
1	1	0	1	1	1	1	1	1
1	1	1	1	0	1	1	0	1
1	0	1	1	0	0	1	0	0
1	0	0	0	0	0	0	0	0

Explanation:

• The above table describes about the 3-bit gray code generator using D-flip flop with the expected gray code inputs d₁, d₂ and d₃ and their calculated values Q=0 and Q’=0.
• If Q=0 and Q’=1 then, the input is 1.
• If Q=1 and Q’=0 then, the input is 0.
• If Q=1 and Q’=1 then, the input is 1.

The above block diagram describes about the 3-bit gray code counter using D-flip flop with the expected gray code inputs d₁, d₂ and d₃. and their calculated values Q=0 and Q’=0.

The logical equation for the inputs d₁, d₂ and d₃ from the above table can be calculated as follows

$\begin{array}{c}{\text{d}}_{\text{1}}\text{=}\overline{{\text{Q}}_{\text{1}}}{\text{Q}}_{\text{2}}\overline{{\text{Q}}_{\text{3}}}\text{+}{\text{Q}}_{\text{1}}{\text{Q}}_{\text{2}}\overline{{\text{Q}}_{\text{3}}}\text{+}{\text{Q}}_{\text{1}}{\text{Q}}_{\text{2}}{\text{Q}}_{\text{3}}\text{+}{\text{Q}}_{\text{1}}\overline{{\text{Q}}_{\text{2}}}{\text{Q}}_{\text{3}}\\ \text{=}{\text{Q}}_{\text{2}}\overline{{\text{Q}}_{\text{3}}}\text{(}\overline{{\text{Q}}_{\text{1}}}\cdot {\text{Q}}_{\text{1}}\text{)}\text{+}{\text{Q}}_{\text{1}}{\text{Q}}_{\text{3}}{\text{(Q}}_{\text{2}}\overline{\cdot {\text{Q}}_{\text{2}}}\text{)}\\ \text{=}{\text{Q}}_{\text{2}}\overline{{\text{Q}}_{\text{3}}}\text{+}{\text{Q}}_{\text{1}}{\text{Q}}_{\text{3}}\end{array}$

$\begin{array}{c}{\text{d}}_2\text{=}\overline{{\text{Q}}_{\text{1}}}\overline{{\text{Q}}_{\text{2}}}{\text{Q}}_{\text{3}}\text{+}\overline{{\text{Q}}_{\text{1}}}{\text{Q}}_{\text{2}}{\text{Q}}_{\text{3}}\text{+}\overline{{\text{Q}}_{\text{1}}}{\text{Q}}_{\text{2}}\overline{{\text{Q}}_{\text{3}}}\text{+}{\text{Q}}_{\text{1}}{\text{Q}}_{\text{2}}\overline{{\text{Q}}_{\text{3}}}\\ \text{=}\overline{{\text{Q}}_{\text{1}}}{\text{Q}}_{\text{3}}\text{(}\stackrel{}{}\end{array}$