Math Bridge: Aliasing and Filter Contracts

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Math BridgeElectronicsStruggle-friendly runway

Why can 80 Hz look exactly like 20 Hz after sampling?

Fold input tones into one sample band and identify what the analogue filter must prevent.

Eddie, the electronics guideEddie guides
The one targetCalculate the observed alias of any input tone at 100 Hz sampling.
The chapter case100 Hz sample rate; 20, 80, 120, and 180 Hz all compared.
What it buys youA release argument for filtering before information becomes ambiguous.

A technician must decide whether alias at 100 samples/s is safe before changing input tone frequency on the real device. The result is unresolved until the rule and units are checked. Predict the direction first.

See the relationship before changing it

The figure reads from left to right. The blue card is input tone frequency. The middle card applies this page's rule. The green card is alias at 100 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 tone frequency, so the numeric fixture does not switch without explanation.

Input tone frequency changes alias at 100 samples/s An input card leads through the rule alias = |tone - 100 x round(tone / 100)| to the alias at 100 samples/s result. INPUT PAGE INPUT APPLY THE RULE predict calculate check units OUTPUT RESULT
Walk the arrows. The sampled alias reflects at Nyquist and at each spectrum copy.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline is 80 Hz.

  2. 2

    Name the relationship. alias = |tone - 100 x round(tone / 100)|

  3. 3

    Substitute with units. |80 - 100 x round(80 / 100)| = 20 Hz

  4. 4

    Read the result. Keep the unit beside the value. Use it only inside the technical boundary on this page.

Predict, then change input tone frequency

Try Predict the direction of alias = |tone - 100 x round(tone / 100)|. Test another input tone frequency, then compare alias at 100 samples/s.

80 Hz
Chapter baseline
Alias at 100 samples/s

Observe The sampled alias reflects at Nyquist and at each spectrum copy. Reset input tone frequency to 80 and compare alias at 100 samples/s.

Explain The sampled alias reflects at Nyquist and at each spectrum copy.

Check yourself

What should you do before trusting a moved-control result?
Answer: Predict its direction, apply the shown relationship, keep the units, and reset to the worked baseline.
What does this small model leave out?
Answer: Only input tone frequency moves here. Field effects named in the technical boundary stay fixed.

Honesty note. This teaching model changes only input tone frequency. It is not a field prediction.

1. Start with the physical story

Sampling keeps only snapshots. A fast wave and a slow wave can cross the same snapshot points, so their stored numbers become identical. A later digital filter receives the ambiguous numbers, not the missing continuous path.

Eddie: The analogue filter protects identity before the ADC discards the evidence needed to tell tones apart.

2. Name every algebra move

1

Invert the sample rateTs=1/fs gives 10.0 ms.

2

Halve the sample ratefN=fs/2 gives 50.0 Hz.

3

Choose the nearest spectrum copyk=round(fin/fs).

4

Subtract and take magnitudefalias=|fin−kfs|.

5

State the boundaryAnything unwanted above Nyquist needs attenuation before sampling.

3. Reproduce the chapter case

fs=100 Hz; Ts=10.0 ms; fN=50.0 Hz
20 Hz: |20−0(100)|=20 Hz
80 Hz: |80−1(100)|=20 Hz
120 Hz: |120−1(100)|=20 Hz
180 Hz: |180−2(100)|=20 Hz

Four different continuous inputs create the same 20 Hz digital trace. The samples alone cannot identify which input existed.

4. Try one real input

TryMove the input tone and watch it reflect at 50 Hz and each 100 Hz spectrum copy.

Input tone
Sample period
Nyquist
Nearest copy order
Observed alias
Samples/input cycle
Alias period
Above Nyquist
Alias share of Nyquist

ObserveThe alias rises from 0 to 50 Hz, then turns back toward zero. At 80, 120, and 180 Hz it returns to 20 Hz through different copy orders.

ExplainSampling repeats the spectrum every 100 Hz. The stored band keeps only distance to the nearest copy, so original frequency identity is lost.

Technical boundaries.

This ledger uses ideal uniform sampling of one steady tone and does not invent a filter absent from the chapter's release inputs.

Filter
Order, cutoff, tolerance, source impedance, load, and required stopband attenuation must be specified.
ADC
Sample-and-hold bandwidth, jitter, aperture, clipping, noise, and quantisation affect captured evidence.
Signal
Real signals contain bands, transients, modulation, noise, and interference rather than one pure tone.

Correct, not complete: this fold ledger does not design or validate an anti-alias filter.

5. Use the result in the design

Define the wanted band and worst unwanted energy, set sampling and transition margin, then choose and measure an analogue filter with enough stopband attenuation before the ADC.

6. Record the evidence state

Record wanted bandwidth, interferers, sample clock and tolerance, filter topology and values, simulated and measured response, ADC input conditions, alias tests, and margin.

7. Check yourself

Why do 80 Hz and 20 Hz both appear at 20 Hz?
Answer: Their difference from the nearest 100 Hz spectrum copy has the same magnitude.
Where does a 50 Hz input land?
Answer: At Nyquist, 50 Hz, where only two samples describe each cycle and phase is fragile.
Can a digital filter recover whether 20 Hz came from 20 or 80 Hz?
Answer: No. After ideal sampling those sources can produce the same sequence; prevention requires analogue filtering or more sampling evidence.
Honesty boundary.

The arithmetic reproduces the chapter's 100 Hz table and keeps filter values symbolic because the release contract has not supplied them.

Filter
Order, cutoff, tolerance, source impedance, load, and required stopband attenuation must be specified.
ADC
Sample-and-hold bandwidth, jitter, aperture, clipping, noise, and quantisation affect captured evidence.
Signal
Real signals contain bands, transients, modulation, noise, and interference rather than one pure tone.

Correct, not complete: this fold ledger does not design or validate an anti-alias filter.

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