Time — Question 8

Question 8

Fine timestamps do not guarantee that all events were recorded

A detector records an incident pulse whenever it is ready. Immediately after each recorded pulse it is blind for $5.0\ \mathrm{ms}$. A pulse arriving during that interval is ignored and does not extend the blind interval. At the end of the interval the detector is ready again. Recorded timestamps have negligible error for this exercise.

1. Determine which input pulses are recorded and which are missed. List the blind intervals and the spacings between recorded timestamps.

2. A student sees only the recorded timestamps and infers a perfectly periodic source. Explain why this inference is not justified. Compare with the spacings in the actual input sequence.

3. Could a source emitting only the recorded pulses produce the same output record? What does this say about reconstructing the number of incident events from these timestamps alone?

4. For this particular five-pulse sequence, what maximum dead time would allow every pulse to be recorded, given that a pulse arriving exactly when the blind interval ends is accepted? Would reducing timestamp digit size alone accomplish this?

Treat all event times and dead times as exact model values. The task concerns lost events, not a statistical rate-estimation formula.

Question 8 - Solution

Fine timestamps do not guarantee that all events were recorded

(a) Update readiness only after a recorded pulse.

$$\mathcal B=(0,5)\cup(7,12)\cup(14,19)\ \mathrm{ms},\qquad t_{\mathrm{out}}=(0,7,14)\ \mathrm{ms}.$$

The pulse at 0 ms is recorded, creating a blind interval from just after 0 up to, but not including, 5 ms. The pulse at 3 ms is missed and does not restart the timer. The pulse at 7 ms is then recorded, making the detector blind until 12 ms. The pulse at 10 ms is missed. The pulse at 14 ms is recorded, followed by blindness until 19 ms.

Thus the output is $\boxed{0,7,14\ \mathrm{ms}}$, with spacings $\boxed{7,7\ \mathrm{ms}}$. The missed input pulses are at 3 and 10 ms.

(b) Periodic output need not imply periodic input.

$$\Delta t_{\mathrm{in}}=(3,4,3,4)\ \mathrm{ms},\qquad\Delta t_{\mathrm{out}}=(7,7)\ \mathrm{ms}.$$

The actual consecutive input spacings are $3,4,3,4\ \mathrm{ms}$, not a constant period. The detector has removed events in a way that makes the remaining record appear periodic. Even an indefinitely repeated alternating-gap input can give this periodic output under the stated rule.

(c) Different inputs can give the same recorded data.

Yes. An input consisting only of pulses at 0, 7, and 14 ms would also produce that output. Hence these timestamps alone do not reveal whether three or five incident pulses occurred; other pulses could also be hidden in blind intervals. A precise time tag on each recorded event does not establish complete event detection.

(d) Compare dead time with the smallest input spacing.

$$\tau_{\max}=\min_j(t_{j+1}-t_j)=3.0\ \mathrm{ms}.$$

The smallest spacing is $3.0\ \mathrm{ms}$. With the stated endpoint convention, any dead time no greater than $\boxed{3.0\ \mathrm{ms}}$ permits this entire sequence to be recorded. This guarantee relies on knowing that sequence or its minimum spacing.

More timestamp digits do not shorten physical recovery time. The detector’s dead time, rather than the precision of its surviving timestamps, is the limitation here.