Math Bridge: Fixed-Point Precision and Battery Days

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Math BridgeEnergy & PowerStruggle-friendly runway

How do fractional bits turn into battery days?

Follow one Q-format choice from its smallest step to rounding noise, ideal SNR, daily charge, and service-life ceilings.

Battery Bruno, the energy and power guideBattery Bruno guides
The one targetChoose fractional bits with both precision and energy consequences visible.
The chapter caseQ7.8 and Q15 plus the chapter's 11.3 versus 0.67 mAh/day workloads.
What it buys youA format decision tied to error and battery budget rather than habit.

A field team faces an unresolved physical question: How do fractional bits turn into battery days? They must answer it before changing fractional 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 fractional bits. The middle card applies this page's relationship. The green card is quantisation step. 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.

Fractional bits changes quantisation step An input card leads through the page relationship to the quantisation step result. SET INPUT ONE CONTROL APPLY RULE predict calculate check units READ RESULT
Walk the arrows. Precision follows the Q-format formula; life follows the measured daily charge. They meet in one decision, but they are not the same calculation.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline for fractional bits is 15.

  2. 2

    Name the relationship. Q7.8: q=2^-8=0.00391, σq=0.00113, SNR=49.9 dB Q15: q=2^-15=0.0000305, σq=0.00000881, SNR=92.1 dB Qusable=0.85x1000=850 mAh 850/11.3=75.2 days; 850/0.67=1269 days

  3. 3

    Substitute the chapter fixture. Set fractional bits to 15. The page ledger gives quantisation step as 3.05e-5 W/Hz.

  4. 4

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

Predict, then change fractional bits

Try Predict the direction of quantisation step. Move one control, calculate, then check your prediction.

15
Chapter baseline
Quantisation step

Observe Precision follows the Q-format formula; life follows the measured daily charge. They meet in one decision, but they are not the same calculation. Reset the control to 15 and compare quantisation step.

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

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

1. Start with the physical story

Fixed-point numbers place an implied binary point inside an integer. F fractional bits divide one unit into 2^F equal steps. A smaller step reduces rounding error. On a processor without a floating-point unit, fixed-point work can also use far fewer cycles, so the format choice reaches the battery ledger.

Battery Bruno: Keep the units beside every number. They show whether a result is charge, energy, voltage, time, or a ratio.

2. Name every algebra move

1

Count the slicesThe step is q=2^−F.

2

Bound typical rounding noiseUniform rounding gives RMS noise q/√12.

3

Express the ideal ratioSNR≈6.02F+1.76 dB.

4

Derate the cellUsable capacity=fderate×Qnominal.

5

Turn charge into daysLife=usable mAh/(mAh per day).

6

Compare compute pathsDivide fixed-point life by floating-point life.

3. Reproduce the chapter case

Q7.8: q=2^−8=0.00391, σq=0.00113, SNR=49.9 dB
Q15: q=2^−15=0.0000305, σq=0.00000881, SNR=92.1 dB
Qusable=0.85×1000=850 mAh
850/11.3=75.2 days; 850/0.67=1269 days

The compute-only fixed-point ceiling is 16.9 times the float ceiling. It is not whole-device life: sensing, radio, sleep current, aging, and self-discharge still spend the same cell.

4. Try one real input

TryMove the control and predict which outputs should change before reading them.

Fractional bits
Quantisation step
RMS rounding noise
Ideal SNR
Usable capacity
Float ceiling
Fixed ceiling
Life ratio
Fixed ceiling

ObserveAdding fractional bits halves the step each time and raises ideal SNR by about 6 dB. The two compute-life numbers stay fixed here because the chapter measured those workloads separately.

ExplainPrecision follows the Q-format formula; life follows the measured daily charge. They meet in one decision, but they are not the same calculation.

Technical boundaries.

This is a small formula ledger, not a complete source qualification.

SNR formula
It describes ideal uniform quantisation, not every fixed-point algorithm.
Capacity
The 850 mAh value is a planning derate, not a measured usable capacity.
Other loads
Radio, sensor, regulator, sleep, and self-discharge are excluded.

Correct, not complete: the bridge connects precision and compute charge; it does not approve a Q-format or battery alone.

5. Use the result in the design

Pick the smallest F that keeps algorithm error inside its tolerance, then measure cycles and current on the target processor. Re-run the battery ledger with every device load included.

6. Record the evidence state

Keep Q-format, scaling, saturation rule, error tolerance, processor, compiler flags, cycle count, active current, workload frequency, and whole-device daily charge.

7. Check yourself

What does one extra fractional bit do to q?
Answer: It halves the quantisation step.
Why is 92.1 dB an ideal limit?
Answer: Real arithmetic also has coefficient, saturation, sensor, and implementation errors.
Is 1,269 days a field-life promise?
Answer: No. It is a compute-only charge ceiling using the stated capacity and derating.
Honesty boundary.

The arithmetic uses the chapter's named or clearly labelled catalog-typical inputs.

SNR formula
It describes ideal uniform quantisation, not every fixed-point algorithm.
Capacity
The 850 mAh value is a planning derate, not a measured usable capacity.
Other loads
Radio, sensor, regulator, sleep, and self-discharge are excluded.

Correct, not complete: the bridge connects precision and compute charge; it does not approve a Q-format or battery alone.

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