Math Bridge: Catch a 90-Second Event Without Aliasing

A field team has a real problem to settle: Catch a 90-Second Event Without Aliasing They must decide what happens before they change logger sample interval on the device. Predict the direction first.

See the relationship first

The figure reads from left to right. The blue card is logger sample interval. The middle card uses this page's rule. The green card is nyquist limit. Follow the arrows: 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.

Follow the arrows. 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.

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?

Answer: Predict its direction, use the shown rule, keep the units, and reset to the worked baseline.

What does this small model leave out?

Answer: Only logger sample interval moves. Field effects named in the page limits stay fixed.

1. Begin with the physical story

Sampling is taking repeated snapshots. If the snapshots are too far apart, a fast repeating change can line up with them in a misleading way and look slower than it really is. That false slow pattern is an alias.

Phoebe: Frequency is events per second, so a 90 s event has a simple rate of 1/90 Hz. Sample frequency is also one divided by the interval. Nyquist requires at least two samples per fastest cycle: f_s ≥ 2f_max.

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 it with every move named

f_s = 1/T_s; f_s ≥ 2f_max; f_alias = |f_signal − n f_s|

Turn duration into frequencyf_max = 1/90 s = 0.0111 Hz.

Apply Nyquistf_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 valuef_alias = |0.0111 − 2(0.005)| = 0.00111 Hz, about a 15-minute false drift.

4. Reproduce the chapter's numbers

The chapter's safe bound is at least one sample every 45 s. A 200 s interval violates it. The 0.0111 Hz event folds 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.

ObserveObserve the Nyquist verdict change and the folded frequency become a slow false pattern when snapshots are too sparse.

ExplainThe 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.

Technical boundaries.

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 calibration fit cannot repair information that sampling already folded into a false frequency. First choose a sample interval that preserves the event, then check ADC resolution, and only then 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. What sample rate does a 90 s event require?

Answer: At least 2/90 = 0.0222 Hz.

2. What is the longest corresponding interval?

Answer: 1/0.0222 ≈ 45 s.

3. Why can a smooth 15-minute drift be dangerous?

Answer: With 200 s sampling, it can be the alias of the 90 s event rather than a real slow temperature change.

Honesty boundary.

These are the chapter inputs, worked results, and named teaching assumptions.

90 s event: Time, interval, or service-life value
200 s logger: Time, interval, or service-life value
12-bit 3.3 V ADC: Voltage or voltage-step value
10 kΩ B = 3950 K thermistor example: Named teaching assumption
reported: Chapter input or worked result
Treating 1/90 Hz as the event's fastest component is a simple planning check: Frequency, sample rate, or event rate
not a complete spectrum: Current or responsivity value

An anti-alias filter, transient shape, sensor time constant, clock jitter, calibration residuals, and held-out validation need the chapter's deeper treatment.