GATE CS 1987 Q13-a — Sequential Circuits

From the given state transitions, we can observe that the circuit is a right-shift register where the bits shift from left to right on each clock pulse:

The input to the first flip-flop $Q_A$ is fed by the output of the $\text{XOR}$ gate. Let us determine the inputs to this $\text{XOR}$ gate by analyzing the transitions of $Q_A$:

• Initial State ($1111$): Next value of $Q_A$ is $0$.
• After 1st clock ($0111$): Next value of $Q_A$ is $0$.
• After 2nd clock ($0011$): Next value of $Q_A$ is $0$.

An $\text{XOR}$ gate connected to $Q_C$ and $Q_D$ perfectly satisfies these conditions:

• At Initial ($1111$): $Q_C \oplus Q_D = 1 \oplus 1 = 0$
• After 1st clock ($0111$): $Q_C \oplus Q_D = 1 \oplus 1 = 0$
• After 2nd clock ($0011$): $Q_C \oplus Q_D = 1 \oplus 1 = 0$

Now, we can find the state after the 4th clock pulse using the state after the 3rd clock ($0001$):

• $Q_A^+ = Q_C \oplus Q_D = 0 \oplus 1 = 1$
• $Q_B^+ = Q_A = 0$
• $Q_C^+ = Q_B = 0$
• $Q_D^+ = Q_C = 0$

Thus, the state $Q_A Q_B Q_C Q_D$ after the fourth clock pulse is $1000$.