Phoebe guidesA field team has a real problem to settle: Can a 200-second sampling gap miss a 90-second transient? They must decide what happens before they change logger sample interval on the device. Predict the direction first.
See the relationship first
Set the input, use the rule, then read the result and its unit.
The audit later on checks more than one number. Here, the added model uses the baseline named below and holds every other chapter value fixed. That sentence bridges the fixtures, so the numbers do not change without a reason.
Derive the baseline in four moves
- 1
Name the input. The chapter baseline for logger sample interval is 200.
- 2
Name the rule. f_s = 1/T_s; f_s ≥ 2f_max; f_alias = |f_signal - n f_s|
- 3
Put in the chapter value. Set logger sample interval to 200. The page rule gives nyquist limit as 0.0025 Hz.
- 4
Read the result. Keep Hz next to the value. Use it only within the limits on this page.
Predict, then change logger sample interval
Try Predict what happens to nyquist limit. Move one control, calculate, then check your idea.
Observe The widget computes f_s = 1/T_s, compares it with 2/90, and applies f_alias = |f_signal - n f_s| from the named derivation moves. Reset to 200 and compare nyquist limit.
Explain Only logger sample interval moves here. The other chapter values stay fixed.
Check yourself
What should you do before you trust the result?
What does this small model leave out?
1. Begin with the physical story
Sampling is taking repeated snapshots. A finite 90 s excursion is a transient, not a repeating cycle with a unique frequency. With snapshots 200 s apart, the entire excursion can fall between samples and go unseen. A different, continuous sinusoid can appear at a lower alias frequency when sampled too slowly.
2. Put names and units on the maths
Keep the units beside every number. They are an error detector: only like units can be added or subtracted.
| Symbol | Meaning | Unit |
|---|---|---|
| f_max | fastest signal component to retain | Hz |
| f_s | samples taken each second | Hz |
| T_s | seconds between samples, 1/f_s | s |
| f_alias | false folded frequency | Hz |
3. Derive the continuous-tone alias with every move named
Name a separate continuous toneChoose a sinusoid at f_signal = 1/90 s = 0.0111 Hz. This is not the finite 90 s excursion.
Apply Nyquist to that tonef_s ≥ 2(0.0111) = 0.0222 Hz.
Invert for intervalT_s ≤ 1/0.0222 = 45 s.
Test the slow loggerAt 200 s, f_s = 1/200 = 0.005 Hz and Nyquist is 0.0025 Hz.
Choose the nearest copyn = 2 puts 2f_s = 0.010 Hz nearest to 0.0111 Hz.
Subtract, then take absolute valueFor the continuous sinusoid, f_alias = |0.0111 − 2(0.005)| = 0.00111 Hz, about a 15-minute apparent cycle.
4. Reproduce the chapter's numbers
A 200 s interval can miss the finite 90 s excursion entirely; it does not give that transient a 15-minute alias. Separately, a continuous 0.0111 Hz sinusoid sampled every 200 s aliases to 0.00111 Hz, whose period is 1/0.00111 ≈ 900 s, or about 15 minutes.
The chapter also checks quantization: a 12-bit 3.3 V ADC has q = 3300/4096 = 0.806 mV, q/√12 = 0.233 mV RMS, and 74.0 dB ideal SNR. Near 0 °C, one code is about 0.0261 °C for the stated thermistor divider.
5. Try the formula
TryMove the logger interval through 45 s and out to the chapter's 200 s case.
ObserveFor the continuous sinusoid, observe the Nyquist verdict and its folded frequency as the sample interval changes. This calculation does not model the finite transient.
ExplainThe widget uses a continuous 0.0111 Hz sinusoid, computes f_s = 1/T_s, compares it with 2f_signal, and applies f_alias = |f_signal − n f_s|. It is a separate example from transient detection.
This small widget varies one named input and holds the chapter constants fixed.
- The honesty boundary below names what it does not model
- Needs separate evidence
Use field evidence or a deeper model before release.
6. What the result buys you
A 200 s gap can miss a finite 90 s event, so transient detection needs an interval chosen for the event and system response. Separately, a continuous sinusoid sampled too slowly can alias to a false frequency. A calibration fit cannot recover either event data that was never sampled or a tone's original frequency after aliasing. Then check ADC resolution and compare linearization residuals. The Under the Hood section handles changing thermistor sensitivity, held-out error, extrapolation, and model failure.
7. Check yourself
Try each question before revealing the answer.
1. For a continuous 0.0111 Hz sinusoid, what sample rate meets the Nyquist bound?
Answer: At least 2(0.0111) = 0.0222 Hz.
2. What is the longest corresponding interval?
Answer: 1/0.0222 ≈ 45 s.
3. What does the 0.00111 Hz alias describe?
Answer: It is the alias of the separate continuous 0.0111 Hz sinusoid sampled every 200 s, not an inevitable appearance of the finite 90 s excursion.
These are the chapter inputs, worked results, and named teaching assumptions.
- 90 s transient duration
- Finite event that may fall between 200 s samples
- 200 s logger interval
- Time between samples
- Continuous 0.0111 Hz sinusoid
- Separate aliasing example
- 12-bit 3.3 V ADC
- Voltage or voltage-step value
- 10 kΩ B = 3950 K thermistor example
- Named teaching assumption
Transient shape, an anti-alias filter, sensor time constant, clock jitter, calibration residuals, and held-out validation need the chapter's deeper treatment.