Phoebe guidesA field team faces an unresolved physical question: Why 55 Hz can masquerade as 45 Hz They must answer it before changing sample interval in seconds on the real device. Predict the direction first.
See the relationship before changing it
The figure reads from left to right. The blue card is sample interval in seconds. The middle card applies this page's relationship. The green card is nyquist ceiling. Walk the arrows once: set the input, apply the rule, then read the result with its unit.
The retained audit below checks several chapter fixtures. This added model holds every other chapter fixture fixed, so the numeric fixture does not switch without explanation.
Derive the baseline in four named moves
- 1
Name the input. The chapter baseline for sample interval in seconds is 0.01.
- 2
Name the relationship. f_s=1/T_s; f_N=f_s/2; f_alias=|f-kf_s|
- 3
Substitute the chapter fixture. Set sample interval in seconds to 0.01. The page ledger gives nyquist ceiling as 50 Hz.
- 4
Read the result. Keep Hz beside the value. Use it only inside the technical boundary on this page.
Predict, then change sample interval in seconds
Try Predict the direction of nyquist ceiling. Move one control, calculate, then check your prediction.
Observe The widget uses the same reciprocal, half-rate, and nearest-copy equations shown above. Reset the control to 0.01 and compare nyquist ceiling.
Explain Only sample interval in seconds moves here. The other chapter fixtures remain fixed.
Check yourself
What should you do before trusting a moved-control result?
What does this small model leave out?
1. Sampling takes timed snapshots
The sample rate f_s is the number of snapshots per second. A repeating wave needs enough snapshots to distinguish its upward and downward parts from a slower impostor.
2. Learn the half-rate bound
Nyquist's condition is f_s ≥ 2f_max. Rearranging by dividing both sides by two gives f_max = f_s/2.
3. See how folding happens
Locate the nearest sample-rate copyFor a 55 Hz tone sampled at 100 Hz, the nearest copy is kf_s = 100 Hz.
Take the absolute gapf_alias = |f − kf_s| = |55−100|.
Name the impostorThe samples look like a 45 Hz tone even though the input was 55 Hz.
4. Try changing the sample interval
TryMove from a 1 ms interval (1,000 samples/s) to the chapter's 10 ms interval (100 samples/s).
ObserveAt 100 samples/s the ceiling is 50 Hz and the 55 Hz input folds to 45 Hz.
ExplainThe widget uses the same reciprocal, half-rate, and nearest-copy equations shown above.
A real low-pass filter rolls off gradually.
- A nominal 50 Hz cutoff still passes energy near 50 Hz, so the chapter's edge-set filter is not a universal production choice
- Needs separate evidence
Use field evidence or a deeper model before release.
5. Work the chapter's two ceilings
The raw 1,000 samples/s stream can represent below 500 Hz if the analog front end already rejects higher content. Decimation to 100 samples/s lowers the limit to 50 Hz. The reported 333× bandwidth reduction is useful for slow trends, but not for preserving every vibration near the edge.
6. Name the engineering fix
Filter before throwing samples away, and choose a cutoff below the new 50 Hz ceiling—often 30–40 Hz here—so the filter has room to roll off. The exact margin depends on filter order and the signal that matters.
7. Check yourself
What is the raw-stream ceiling?
What is the decimated ceiling?
Where does 55 Hz fold at 100 samples/s?
These are the chapter inputs, worked results, and named teaching assumptions.
- 1,000
- Chapter input or worked result
- 100 samples/s rates
- Frequency, sample rate, or event rate
- 500
- Chapter input or worked result
- 50 Hz limits
- Frequency, sample rate, or event rate
- 50 Hz filter
- Frequency, sample rate, or event rate
- 333× goal come from the chapter
- Percentage, ratio, or gain
The 55-to-45 Hz example exposes the edge risk; it does not model a particular filter's measured response.