GATE Electrical Engineering (EE) 2021 Solved Papers

Question : 63 of 65

Marks: +1, -0

A counter is constructed with three D flip-flops. The input-output pairs are named

( {D}_{0}, {Q}_{0}),( {D}_{1}, {Q}_{1})

and

( {D}_{2}, {Q}_{2})

, where the subscript 0 denotes the least significant bit. The output sequence is desired to be the Gray-code sequence

000,001,011,010,110,111,101

, and 100 , repeating periodically. Note that the bits are listed in the

{Q}_{2} {Q}_{1} {Q}_{0}

format. The combinational logic expression for

{D}_{1}

is:

Solution:

$\Q_2$	$\Q_1$	$\Q_0$	$\Q_2^{+}$	$\Q_1^{+}$	$\Q_0^{+}$	$\D_2$	$\D_1$	$\D_0$
0	0	0	0	0	1	0	0	1
0	0	1	0	1	1	0	1	1
0	1	0	1	1	0	1	1	0
0	1	1	0	1	0	0	1	0
1	0	0	0	0	0	0	0	0
1	0	1	1	0	0	1	0	0
1	1	0	1	1	1	1	1	1
1	1	1	1	0	1	1	0	1

For

D_1

D_{1}=\ov{Q}_{2} Q_{0}+Q_{1} \ov{Q_{0}}

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$\Q_2$	$\Q_1$	$\Q_0$	$\Q_2^{+}$	$\Q_1^{+}$	$\Q_0^{+}$	$\D_2$	$\D_1$	$\D_0$
0	0	0	0	0	1	0	0	1
0	0	1	0	1	1	0	1	1
0	1	0	1	1	0	1	1	0
0	1	1	0	1	0	0	1	0
1	0	0	0	0	0	0	0	0
1	0	1	1	0	0	1	0	0
1	1	0	1	1	1	1	1	1
1	1	1	1	0	1	1	0	1

$\Q_2$	$\Q_1$	$\Q_0$	$\Q_2^{+}$	$\Q_1^{+}$	$\Q_0^{+}$	$\D_2$	$\D_1$	$\D_0$
0	0	0	0	0	1	0	0	1
0	0	1	0	1	1	0	1	1
0	1	0	1	1	0	1	1	0
0	1	1	0	1	0	0	1	0
1	0	0	0	0	0	0	0	0
1	0	1	1	0	0	1	0	0
1	1	0	1	1	1	1	1	1
1	1	1	1	0	1	1	0	1

$\Q_2$	$\Q_1$	$\Q_0$	$\Q_2^{+}$	$\Q_1^{+}$	$\Q_0^{+}$	$\D_2$	$\D_1$	$\D_0$
0	0	0	0	0	1	0	0	1
0	0	1	0	1	1	0	1	1
0	1	0	1	1	0	1	1	0
0	1	1	0	1	0	0	1	0
1	0	0	0	0	0	0	0	0
1	0	1	1	0	0	1	0	0
1	1	0	1	1	1	1	1	1
1	1	1	1	0	1	1	0	1