Phoebe guides this bridgeSee the relationship before changing it
The figure reads from left to right. The blue card is input signal frequency. The middle card applies this page's rule. The green card is observed alias at 50 samples/s. 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 model keeps those stated values fixed and changes only input signal frequency, so the numeric fixture does not switch without explanation.
Derive the baseline in four named moves
- 1
Name the input. The chapter baseline is 38 Hz.
- 2
Name the relationship. alias = |input - 50 x round(input / 50)|
- 3
Substitute with units. |38 - 50 x round(38 / 50)| = 12 Hz
- 4
Read the result. Keep the unit beside the value. Use it only inside the technical boundary on this page.
Predict, then change input signal frequency
Try Predict the direction of alias = |input - 50 x round(input / 50)|. Test another input signal frequency, then compare observed alias at 50 samples/s.
Observe A tone above Nyquist can fold into band before link loss is considered. Reset input signal frequency to 38 and compare observed alias at 50 samples/s.
Explain A tone above Nyquist can fold into band before link loss is considered.
Check yourself
What should you do before trusting a moved-control result?
What does this small model leave out?
1. Start with what a sample is
A continuous signal exists between measurements. Sampling keeps only one value at each clock tick. A frequency says how many cycles happen each second; a 10 Hz sampler keeps ten values per second.
Phoebe: A sample is evidence from an instant, not a photograph of everything that happened between instants.
2. Name the highest safe frequency
To distinguish the rising and falling parts of a wave, the sampler needs at least two samples per cycle. This is the Nyquist boundary.
State the boundaryfs ≥ 2fmax
Divide both sides by 2fmax ≤ fs/2
Substitute 10 Hzfmax ≤ 10/2 = 5.0 Hz
3. Fold the 60 Hz ripple
Sampling repeats the spectrum every fs. Choose the integer k that brings the component into the 0-to-5 Hz observed band.
Choose the nearest repeated copy60/10 = 6, so k = 6.
Substitute|60 − 6(10)| Hz
Multiply inside the brackets|60 − 60| Hz
Subtract, then take absolute value|0| = 0.0 Hz
The ripple now looks exactly like DC. A digital filter cannot separate two causes that produced the same stored samples.
4. Contrast bounded quantization
A 10-bit ADC has 210 = 1024 codes. Its 5 V range is 5000 mV.
Use the code-step ruleq = Vref/2N
Substitute chapter valuesq = 5000/1024 mV
Divideq = 4.88 mV
Substitute bits into ideal SNR6.02(10)+1.76 = 62.0 dB
This rounding error is bounded and knowable; unlike aliasing, it does not invent an indistinguishable frequency.
5. Work the recoverable radio boundary
The chapter starts its 915 MHz model at 31.7 dB loss over 1 m, then uses path exponent n = 2.8 over 800 m.
Apply the distance term31.7 + 10(2.8)log10(800)
Evaluate the logarithm and addPath loss ≈ 113 dB.
Form the marginM = EIRP − loss − sensitivity
Substitute signed valuesM = 20 − 113 − (−110) = 17.0 dB
Add recoverable antenna gain17.0 + 6 = 23.0 dB
6. Check yourself
1. What is the Nyquist ceiling at 10 Hz?
5.0 Hz, because 10/2 = 5.
2. Why can’t a later filter remove this 60 Hz ripple?
At 10 Hz sampling it aliases to 0.0 Hz, exactly where the wanted DC value sits.
3. What does a 6 dBi antenna do to 17 dB margin in this model?
It raises the recoverable margin to 23.0 dB, provided placement and regulation allow it.
7. Honesty boundary
These are the chapter inputs, worked results, and named teaching assumptions.
- 60 Hz/10 Hz collision is a deliberately exact alias example
- Named teaching assumption
Real signals have phase, finite observation windows, analog noise, filter roll-off, clock jitter, and multiple spectral components. The 915 MHz path model uses the chapter’s catalog-typical 31.7 dB reference and n = 2.8; a site survey must replace that estimate. Go deeper in the chapter’s Under the Hood section for boundary-by-boundary loss and delivery evidence.