The following figure shows a sequential circuit with three D Flip Flops with a common input clock. Assume that the flip flops are initialized at “1”. That is,
during Cycle 0. This means the output of this circuit is initially
. a. Compute the value of each output signal (
,
and
) for 8 cycles using the table below. b. Convert the 3-bit binary
to its equivalent decimal in the table. How many unique output states (i.e.,
) does this circuit produce?
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D_2D_1D_0Q_2Q_1Q_0
Q = Q_2Q_1Q_0$) 0 1 1 1 7 1 2 3 4 5 6 7 8
The Correct Answer and Explanation is:
This problem requires analyzing a synchronous sequential circuit composed of three D flip-flops and an XOR gate. The goal is to trace the circuit’s output states over eight clock cycles and determine the number of unique states it produces.
a. Completed Table
The state of the circuit at the beginning of any clock cycle t+1 is determined by the D inputs calculated from the state at clock cycle t. The input equations for the flip-flops are:
- D₂ = Q₂ ⊕ Q₁ (The output of the XOR gate with inputs Q₂ and Q₁)
- D₁ = Q₂
- D₀ = Q₁
Using these equations, we can fill the table starting from the initial state Q₂Q₁Q₀ = 111 at Cycle 0.
| Clock Cycle | D₂ | D₁ | D₀ | Q₂ | Q₁ | Q₀ | Decimal (Q=Q₂Q₁Q₀) |
| 0 | 0 | 1 | 1 | 1 | 1 | 1 | 7 |
| 1 | 1 | 0 | 1 | 0 | 1 | 1 | 3 |
| 2 | 1 | 1 | 0 | 1 | 0 | 1 | 5 |
| 3 | 0 | 1 | 1 | 1 | 1 | 0 | 6 |
| 4 | 1 | 0 | 1 | 0 | 1 | 1 | 3 |
| 5 | 1 | 1 | 0 | 1 | 0 | 1 | 5 |
| 6 | 0 | 1 | 1 | 1 | 1 | 0 | 6 |
| 7 | 1 | 0 | 1 | 0 | 1 | 1 | 3 |
| 8 | 1 | 1 | 0 | 1 | 0 | 1 | 5 |
b. Unique Output States
By observing the “Decimal” column in the table above, we can identify the sequence of output states produced by the circuit. The sequence is: 7, 3, 5, 6, 3, 5, 6, 3, 5, …
After the initial state of 7, the circuit enters a repeating cycle of 3 → 5 → 6. To find the total number of unique states the circuit produces starting from its initial condition, we list all the distinct decimal values that appear in the sequence.
The unique states are 7, 3, 5, and 6.
Therefore, this circuit produces 4 unique output states.thumb_upthumb_down
