You are given a free running clock with a duty cycle of 50% and a digital…

Answer: The topology that delays the waveform by 180° (half a clock period) is the circuit that uses two D flip-flops clocked on complementary edges without inverting the data path (the circuit shown in the image labeled C).

Key idea:

• Use two edge-triggered storage stages that sample on opposite (complementary) clock edges so that the value is transferred from the first stage to the second on the next (complementary) edge. That transfer introduces a half-clock-period latency.

• Avoid inserting extra inverters in the data path or using complemented outputs as those change polarity or timing and prevent getting a clean 180° phase shift.

Step-by-step reasoning (timing perspective):

• Because the input f only changes at one clock edge, sampling it on the complementary edge ensures the sampled value is stable when captured.

• The first flip-flop captures the current value of f on one edge; the second flip-flop, clocked on the opposite edge, captures the first stage’s output on the next edge. The gap between those two edges is half a clock period, so the value appears at the final output delayed by half a period (180°).

• If either stage inverts the signal or both stages sample on the same effective edge, the result is not the desired half-period delayed replica.

Conclusion: Use the chain of two D flip-flops clocked on complementary edges (no data inversion) to obtain the required 180° phase delay. This matches the circuit shown in the provided diagram C.