A clocked sequential circuit using D flip-flops must repeatedly follow the…

Concept

For a D flip-flop, the input D is the bit’s next-state value at the active clock edge. An n-bit state register can encode 2n states. In a controlled binary up-counter, the least significant bit toggles whenever counting is enabled, while the next bit toggles only when counting is enabled and the current least significant bit is 1.

Application

Let Q_1 be the most significant bit and Q_0 the least significant bit. The required next states are:

1. There are four states, so the minimum register size is ⌈log2 4⌉ = 2 flip-flops.

2. When x = 0, both bits hold their values. When x = 1, Q_0 toggles on every clock. Therefore D_0 = Q_0 XOR x.

3. The bit Q_1 toggles only when counting is enabled and Q_0 = 1. Therefore D_1 = Q_1 XOR (x AND Q_0).

4. The specification defines the state bits and the control input x; it does not define a separate external output function.

Cross-check and contrast

• Three flip-flops provide eight encodings, although the sequence contains four states.

• One external output is not implied because no output function is specified.

• The MSB input expression includes x through x AND Q_0.

• The LSB input expression includes x directly through Q_0 XOR x.

Therefore, the correct statement is: the input of the rightmost flip-flop, which stores the least significant state bit, depends on the external input x.