A sampler only sees certain instants. Change the signal and the sampling rate to discover how different curves can leave the same observations.
The model: an ideal, unit-amplitude, zero-phase cosine, uniformly sampled over one second. The two curves are authored signals; the dots are their shared observations. No sound, hardware or uploaded data is involved.
Displayed numbers round to six decimal places. The CSV keeps full numeric precision. Tiny floating-point differences can occur when evaluating cosine.
| Index n | Time (s) | Reference | Alternative |
|---|
Predict a result before changing the controls.
At times t = n / fs, adding one entire sample rate to a cosine’s frequency adds 2πn to its phase. Cosine repeats, so the observations agree for every integer n.
The 9 Hz reference has the same samples as 1 Hz. Change the reference to 1 Hz: the lab now shows 9 Hz as a different matching alternative. More samples at the same integer times cannot distinguish this pair.
With a known band limit B, the ideal strict condition is fs > 2B. It concerns a complete ideal sample sequence. A finite plot does not establish the band limit or prove practical recovery.
That signal varies between samples yet looks identical to zero at every sample time. The lab uses a zero-phase cosine, which instead alternates +1 and −1 at this boundary. One specified phase does not justify arbitrary-phase recovery.
Unwanted high-frequency content can fold into the band you want. An appropriate anti-alias filter attenuates it before sampling. A later filter cannot distinguish causes that already gave identical readings.
For a new acquisition of the authored 9 Hz reference, 24 samples/sec puts 9 Hz below the 12 Hz half-rate. It does not repair an older recording. This lab does not design a physical filter or test real equipment.
Take the original sampling course into RecallWeave’s native deck format. It includes explanations and transfer prompts on the clock, alias families, recovery assumptions, and acquisition choices. The adjacent companion guide contains worked answers. Deck import availability depends on your RecallWeave version; the experiment above works directly from this file.
Original AI-assisted course wording, worked examples and field-note text: CC0-1.0. The linked MIT materials retain their own licenses; their exercises, prose and figures are not reproduced. Other repository content is not relicensed. This lesson makes no claim about learning efficacy, unknown physical sources, finite-record reconstruction, or real acquisition safety.