Phoebe guidesA field team has a real problem to settle: Record Length and Frequency Resolution They must decide what happens before they change record duration on the device. Predict the direction first.
See the relationship first
The figure reads from left to right. The blue card is record duration. The middle card uses this page's rule. The green card is frequency spacing. 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.
Derive the baseline in four moves
- 1
Name the input. The chapter baseline for record duration is 1.
- 2
Name the rule. T = N/f_s; Δf = f_s/N = 1/T
- 3
Put in the chapter value. Set record duration to 1. The page rule gives frequency spacing as 1.00 Hz.
- 4
Read the result. Keep Hz next to the value. Use it only within the limits on this page.
Predict, then change record duration
Try Predict what happens to frequency spacing. Move one control, calculate, then check your idea.
Observe The outputs use N = f_sT and Δf = 1/T, the same equations derived above. Reset to 1 and compare frequency spacing.
Explain Only record duration 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. Two knobs answer two questions
Sample rate asks, “How often did I look?” Record length asks, “How long did I keep looking?” A high rate helps avoid aliasing of fast signals. A long record helps distinguish slow beat patterns between nearby frequencies.
2. Count samples across time
If f_s samples arrive each second and the capture contains N samples, then its duration is N divided by f_s. A discrete spectrum has N frequency slots spread across f_s hertz, so the spacing is f_s/N.
| Symbol | Meaning | Unit |
|---|---|---|
| f_s | sample rate | samples per second (Hz) |
| N | samples kept | samples |
| T | record duration | seconds |
| Δf | frequency-bin spacing | hertz |
3. Derive the spacing
Start with sample ratef_s = N/T: samples divided by seconds.
Make time the subjectMultiply by T and divide by f_s to get T = N/f_s.
Write spectrum spacingN bins across f_s hertz gives Δf = f_s/N.
Substitute N = f_sTΔf = f_s/(f_sT) = 1/T.
4. Reproduce Exercise 5
At 1 kHz, N = 1,000 samples lasts T = 1,000/1,000 = 1.00 s, so Δf = 1/T = 1.00 Hz. That can separate the chapter's 50 Hz and 53 Hz tones, which are 3 Hz apart. It cannot separate tones only 0.5 Hz apart. For that, T ≥ 1/0.5 = 2.00 s and N = 1,000×2 = 2,000 samples.
5. Try the same formula
TryStretch the 1 kHz record from 0.5 s to 4 s and find when a 0.5 Hz tone gap becomes resolvable by the simple bin-spacing check.
ObserveDoubling time from 1 s to 2 s doubles N and halves the spacing from 1.00 Hz to 0.50 Hz.
ExplainThe outputs use N = f_sT and Δf = 1/T, the same equations derived above.
The simple “gap at least Δf” check assumes a clean finite record and does not model window shape, leakage, noise, or estimator choice.
- No extra effects are represented beyond the stated model
- Needs separate evidence
Use field evidence or a deeper model before release.
6. What the result buys you
Write sample rate and record length separately in the lab record. Raising f_s without raising N can shorten T and make frequency spacing worse. Lengthening T cannot repair aliasing if f_s was already too low. Both checks must pass.
7. Check yourself
1. How long are 1,000 samples at 1 kHz?
Answer: T = 1000/1000 = 1.00 s.
2. What spacing does a two-second record provide?
Answer: Δf = 1/2 = 0.50 Hz.
3. Does a faster sample rate alone guarantee finer spacing?
Answer: No. If N does not grow too, the record can become shorter. Spacing depends on T.
These are the chapter inputs, worked results, and named teaching assumptions.
- 1 kHz
- Frequency, sample rate, or event rate
- 1,000-sample
- Device, payload, or sample count
- 1.00 s
- Time, interval, or service-life value
- 1.00 Hz
- Frequency, sample rate, or event rate
- 50/53 Hz
- Frequency, sample rate, or event rate
- 0.5 Hz
- Frequency, sample rate, or event rate
- 2.00 s
- Time, interval, or service-life value
- 2,000-sample cases reproduce the companion lab. Δf = 1/T is the basic rectangular-record spacing
- Capacitance value
- not a guarantee that any two noisy tones will be visibly separated
- Current or responsivity value
- windowing
- Chapter input or worked result
- leakage
- Chapter input or worked result
- signal-to-noise ratio
- Percentage, ratio, or gain
- the lab's separate Nyquist check still matter
- Time, interval, or service-life value
Treat these figures as teaching evidence, not as a complete release claim.