A sensor retains selected information about a changing signal. Its sampling times, exposure window and response law determine what survives. Two different continuous signals can therefore produce the same sequence of readings, even when every recorded number is exact.
Two signals, one record
Consider unit-amplitude signals x(t) = cos(2π × 2t) and z(t) = cos(2π × 10t), with t measured in seconds. One oscillates at 2 hertz and the other at 10 hertz. Sample both instantaneously twelve times per second, at times tₙ = n/12.
At those times, the second signal has phase 2πn − 2π(2n/12). The cosine is unchanged by a whole-number turn or by reversing the sign of its angle. Therefore z(tₙ) = x(tₙ) for every integer n.
The ambiguity is exact. A more precise digitizer at the same sampling times would not distinguish these signals. Different sampling or an independently justified restriction on the possible signal frequencies can provide the missing information.
Inspect the samples
Change the sampling rate or the duration of a centred exposure. Curves show the two underlying signals; dots show the retained readings. Both signals are defined beyond the plotted half-second, so a centred exposure at an endpoint remains well defined.
12 samples/s; instantaneous sampling; 7 readings per signal in the plotted interval. Maximum difference between corresponding readings: 0.000. The readings coincide.
Teal curves: underlying unit-amplitude signals. Brown dots: recorded values. This is a noiseless teaching model; it does not simulate a particular camera or detector.
Exposure is an operation
A finite exposure need not record the signal at a single instant. In this example, it records the average over an interval of duration Δ centred on the timestamp. Integrating a cosine over that interval gives
The factor tends to one as Δ approaches zero. For a 10 Hz signal and Δ = 0.10 seconds, the window spans a complete cycle and the average is zero. For the 2 Hz signal at the same exposure, the factor is approximately 0.9355.
The differing readings at finite exposure use the known equal input amplitudes in this comparison. With unknown source amplitudes, an amplitude change can mimic part of the exposure effect. The observation model must state which quantities are known and which are to be inferred.
Sampling and reconstruction
Shannon's 1949 paper gives a foundational account of reconstructing a suitably band-limited signal from regularly spaced samples. The bandwidth restriction is an essential hypothesis. For robust use, the sampling rate must exceed twice the highest frequency admitted in the signal model, with the measurement and filtering assumptions also specified.
The demonstration changes to 25 samples per second to distinguish the displayed 2 Hz and 10 Hz signals. That successful comparison does not prove that a finite set of samples uniquely reconstructs every imaginable continuous signal. Noise, finite observation time and departures from the assumed bandwidth require further treatment.
What the example says about time
Discrete readings can arise from continuous dynamics through an instrument's sampling procedure. They therefore do not, by themselves, establish that time is fundamentally discrete. Conversely, a smooth-looking reconstruction does not prove that its underlying ontology is continuous. Each inference requires a model and distinguishing evidence.
For SMM/DTT, the useful connection is the explicit observation map: which distinctions survive outer completion, and which further measurements could reveal differences between underlying histories? The memory essay asks what those retained distinctions must contain for prediction.
Sources and further reading
- C. E. Shannon, “Communication in the Presence of Noise,” Proceedings of the IRE 37 (1949), 10–21, section II. Institutional copy of the published paper. The two-frequency identity and exposure integral are worked explicitly here.
- Time inside a simulation; what counts as an observer.
- The timing programme's experimental pathway; Single Monad foundations and book connections.