Math Bridge: ADC Noise Shaping

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Math BridgeElectronicsStruggle-friendly runway

Why does loop order change the value of doubling the sample rate?

Connect noise shaping and OSR octaves to in-band decibels and ideal extra bits.

Eddie, the electronics guideEddie guides
The one targetTurn loop order and OSR into an ideal in-band noise-improvement ledger.
The chapter caseFirst-order shaping at OSR 64, then OSR 128.
What it buys youA reason for the 3, 9, and 15 dB-per-octave rules.

A technician must decide whether ideal gain per osr doubling is safe before changing noise-shaping loop order 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 noise-shaping loop order. The middle card applies this page's rule. The green card is ideal gain per osr doubling. 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 noise-shaping loop order, so the numeric fixture does not switch without explanation.

Noise-shaping loop order changes ideal gain per osr doubling An input card leads through the rule gain = (2 x order + 1) x 3.0103 dB to the ideal gain per osr doubling result. INPUT PAGE INPUT APPLY THE RULE predict calculate check units OUTPUT RESULT
Walk the arrows. Higher loop order makes each oversampling octave worth more ideal noise reduction.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline is 1.

  2. 2

    Name the relationship. gain = (2 x order + 1) x 3.0103 dB

  3. 3

    Substitute with units. (2 x 1 + 1) x 3.0103 = 9.031 dB

  4. 4

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

Predict, then change noise-shaping loop order

Try Predict the direction of gain = (2 x order + 1) x 3.0103 dB. Test another noise-shaping loop order, then compare ideal gain per osr doubling.

1
Chapter baseline
Ideal gain per OSR doubling

Observe Higher loop order makes each oversampling octave worth more ideal noise reduction. Reset noise-shaping loop order to 1 and compare ideal gain per osr doubling.

Explain Higher loop order makes each oversampling octave worth more ideal noise reduction.

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 noise-shaping loop order moves here. Field effects named in the technical boundary stay fixed.

Honesty note. This teaching model changes only noise-shaping loop order. It is not a field prediction.

1. Start with the physical story

Oversampling narrows the useful frequency band. Feedback can also push quantisation noise upward in frequency, where a decimation filter removes it. Higher loop order makes the in-band tail fall faster.

Eddie: The converter does not create free information; it trades sample rate, feedback order, and filtering for less noise in a chosen band.

2. Name every algebra move

1

Name loop orderAn order-L shaper gives an in-band power exponent 2L+1.

2

Price one octaveDoubling OSR divides noise power by 2^(2L+1).

3

Convert to decibelsUse 10 log10 of the power ratio.

4

Count octavesUse log2(OSR).

5

Estimate ideal bitsDivide the SNR gain by 6.02 dB per bit.

3. Reproduce the chapter case

L=1 gives 2L+1=3
gain per octave=3(10 log10 2)=9.031 dB
OSR 64=2^6, so gain=6(9.031)=54.2 dB
ideal extra bits=54.2/6.02=9.00
OSR 128 adds 9.031 dB, or about 1.50 bits

This is the ideal slope behind the chapter's first-order OSR 64 and 128 comparisons.

4. Try one real input

TryChange loop order and watch one OSR octave become more valuable.

Loop order
Power exponent
Gain per octave
Bits per octave
Octaves at OSR 64
Gain at OSR 64
Ideal bits at OSR 64
Doubled OSR
Gain at doubled OSR
Ideal bits at doubled OSR

ObserveEach extra loop order adds two powers to the OSR law, so the ideal dB gain per doubling rises by about 6.02 dB.

ExplainHigher order steepens the noise-transfer function near zero frequency; it does not simply average more copies of the same sample.

Technical boundaries.

This ledger exposes an ideal low-frequency noise-shaping law.

Idealisation
It assumes white quantisation noise, a stable loop, and a suitable decimation filter.
Converter
Thermal noise, jitter, idle tones, overload, mismatch, and reference noise set real floors.
Bits
SNR-derived effective bits are not guaranteed monotonic accuracy or usable output bits.

Correct, not complete: this ledger does not design, stabilise, or qualify a sigma-delta ADC or its filter.

5. Use the result in the design

Use the ideal slope as a ceiling, then compare it with the converter's measured SNR, bandwidth, latency, stability, and filter rejection.

6. Record the evidence state

Record loop order, modulator rate, signal bandwidth, OSR definition, decimation filter, input level, clock source, SNR method, and observed tones.

7. Check yourself

Why does plain oversampling use L=0?
Answer: With no shaping zero, the exponent is one and each OSR doubling ideally gains only 3.01 dB of in-band noise power.
Why is 9.00 extra bits not a guaranteed resolution claim?
Answer: It is an ideal SNR conversion; real noise, distortion, stability, and linearity limit usable resolution.
What does doubling OSR from 64 to 128 add at first order?
Answer: One octave, ideally 9.03 dB or about 1.50 bits.
Honesty boundary.

The arithmetic reproduces the chapter's ideal OSR cases; it is not an ADC performance guarantee.

Idealisation
It assumes white quantisation noise, a stable loop, and a suitable decimation filter.
Converter
Thermal noise, jitter, idle tones, overload, mismatch, and reference noise set real floors.
Bits
SNR-derived effective bits are not guaranteed monotonic accuracy or usable output bits.

Correct, not complete: this ledger does not design, stabilise, or qualify a sigma-delta ADC or its filter.

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