Math Bridge: Bit depth and radio cost

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Math BridgeAnalytics & MLStruggle-friendly runway

Why does cutting bit depth save less than extracting the right edge feature?

Compare converter evidence with raw-radio cost.

Data Dora, the guideData Dora guides
The one targetCalculate the trade between bits and radio work.
The chapter case1 kHz; 16 bit; 500 sensors; 95% feature reduction.
What it buys youA defensible edge-compression choice.

A field team faces an unresolved physical question: Why does cutting bit depth save less than extracting the right edge feature? They must answer it before changing bits 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 bits. The middle card applies this page's relationship. The green card is raw bit/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 added model holds every other chapter fixture fixed, so the numeric fixture does not switch without explanation.

Bits changes raw bit/s An input card leads through the page relationship to the raw bit/s result. SET INPUT ONE CONTROL APPLY RULE predict calculate check units READ RESULT
Walk the arrows. The stated 95% feature reduction saves much more radio work without redefining input resolution.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline for bits is 16.

  2. 2

    Name the relationship. 1,000x16 = 16.0 kbps = 2,000 B/s 6.02(16)+1.76 = 98.1 dB 16,000x50 nJ = 0.800 mW; after 95% reduction = 0.040 mW

  3. 3

    Substitute the chapter fixture. Set bits to 16. The page ledger gives raw bit/s as 16000.

  4. 4

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

Predict, then change bits

Try Predict the direction of raw bit/s. Move one control, calculate, then check your prediction.

16
Chapter baseline
Raw bit/s

Observe The stated 95% feature reduction saves much more radio work without redefining input resolution. Reset the control to 16 and compare raw bit/s.

Explain Only bits moves here. The other chapter fixtures remain fixed.

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 bits moves. Field effects named in the page's technical boundary stay fixed.

Honesty note. This teaching model changes only bits. It replays the chapter fixture; it is not a field prediction.

1. Separate two promises

Sample rate protects frequency content. Bit depth controls ideal amplitude resolution. Both affect payload, but they are not interchangeable.

Data Dora: Throwing away amplitude bits is not the same as selecting a useful feature.

2. Name every algebra move

1

MultiplyR=fsN.

2

Convert to bytesB/s=R/8.

3

Estimate ideal rangeSNR=6.02N+1.76.

4

Price each bitP=REbit.

3. Reproduce the chapter

1,000×16 = 16.0 kbps = 2,000 B/s
6.02(16)+1.76 = 98.1 dB
16,000×50 nJ = 0.800 mW; after 95% reduction = 0.040 mW

Eight bits halve the payload but drop the ideal ceiling to 49.9 dB.

4. Try the bit depth

TryCompare 8–16 bits without changing the sample rate.

Bits
Raw bit/s
Raw bit rate
Raw bytes
Ideal SNR
Radio power
Fleet radio power
Feature bit rate
Feature radio power

ObserveEach lost bit saves 6.25% of the 16-bit payload but costs about 6.02 dB.

ExplainThe stated 95% feature reduction saves much more radio work without redefining input resolution.

Technical boundaries.

Radio power scales ideally with bits here.

Converter
No analogue noise or effective-number-of-bits loss
Radio
No headers, retries, idle, startup, or coding overhead
Features
No claim that FFT peaks preserve every fault

Measure end-to-end detection and device energy.

5. Preserve useful evidence

Choose features against named fault signatures, then retain raw anomaly windows for audit and retraining.

6. Version the trade

Record sample rate, bit depth, analogue range, feature recipe, payload schema, radio mode, and energy measurement.

7. Check yourself

What is the 16-bit payload?
Answer: 16 kbps or 2,000 B/s.
What does 8 bit cost?
Answer: About 48.2 dB of ideal dynamic range versus 16 bit.
Does 95% reduction prove fault retention?
Answer: No; it must be validated against labeled events.
Honesty boundary.

The stream and reduction come from the chapter; energy per bit is a stated typical assumption.

1 kHz, 16 bit, 500 sensors
Chapter scenario
95%
Chapter FFT reduction
50 nJ/bit
Teaching radio assumption

Correct, not complete: a bit ledger does not qualify an edge fault detector.

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