Math Bridge: LoRa Spreading Factor, Rate, and Gain

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Math BridgeLoRaWANSpreading trade

What does a higher spreading factor buy, and what does it cost?

Follow chirp states into symbol time, coded bit rate, processing gain, and a capacity boundary.

Eddie, the electronics guideEddie guides
The one targetExplain the SF7-to-SF12 trade with one formula chain.
The chapter case125 kHz, coding fraction 4/5, and a −20 dB SNR screen.
What it buys youAn ADR discussion that keeps margin and airtime together.

A field team faces an unresolved physical question: What does a higher spreading factor buy, and what does it cost? They must answer it before changing spreading factor 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 spreading factor. The middle card applies this page's relationship. The green card is chirp states. 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.

Spreading factor changes chirp states An input card leads through the page relationship to the chirp states result. SET INPUT ONE CONTROL APPLY RULE predict calculate check units READ RESULT
Walk the arrows. The receiver gets more observations per symbol; the link borrows margin from airtime and capacity rather than creating power.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline for spreading factor is 12.

  2. 2

    Name the relationship. Tsym = 2^SF/125000 Rb = SF x (4/5) x 125000/2^SF PG = 10 log10(2^SF) C = 125000 log2(1 + 10^(-20/10))

  3. 3

    Substitute the chapter fixture. Set spreading factor to 12. The page ledger gives chirp states as 4096.

  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 spreading factor

Try Predict the direction of chirp states. Move one control, calculate, then check your prediction.

12
Chapter baseline
Chirp states

Observe The receiver gets more observations per symbol; the link borrows margin from airtime and capacity rather than creating power. Reset the control to 12 and compare chirp states.

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

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

1. Start with the physical story

Spreading factor says how many chirp states represent a symbol. More states let the receiver integrate longer and gain processing margin, but each symbol occupies more time and the coded bit rate falls.

Eddie: A longer look can reveal a weaker pattern, but it also keeps the shared channel busy longer.

2. Name every algebra move

1

Count statesRaise two to the spreading factor.

2

Find symbol timeDivide states by bandwidth.

3

Find coded rateMultiply SF and 4/5 by bandwidth, then divide by states.

4

Find ideal gainTake 10 log10 of the state count and compare with SF7.

3. Reproduce the chapter case

Tsym = 2^SF/125000
Rb = SF × (4/5) × 125000/2^SF
PG = 10 log10(2^SF)
C = 125000 log2(1 + 10^(−20/10))

SF7 gives 1.024 ms, about 5,469 bps, and 21.07 dB. SF12 gives 32.768 ms, about 293 bps, and 36.12 dB—15.05 dB more ideal gain, but 18.67 times slower.

4. Try one real input

TryMove spreading factor while bandwidth, coding fraction, and the −20 dB capacity screen stay fixed.

Spreading factor
Chirp states
Symbol time
Coded bit rate
Ideal processing gain
Gain over SF7
Slowdown versus SF7
Shannon ceiling
Capacity headroom

ObserveEach SF step doubles states and symbol time, while processing gain rises by about 3.01 dB and coded rate falls.

ExplainThe receiver gets more observations per symbol; the link borrows margin from airtime and capacity rather than creating power.

Technical boundaries.

This is an ideal teaching ledger, not a radio data sheet.

Rate
Headers, preamble, low-data-rate optimization, payload, coding, and retries change packet airtime.
Gain
Real sensitivity steps depend on bandwidth, implementation, noise figure, and demodulator behavior.
Capacity
Shannon is a ceiling, not a promised LoRa throughput.

Correct, not complete: use the measured state named above before release.

5. Use the result in ADR review

Select the lowest spreading factor that keeps measured margin and delivery within policy, then check airtime, duty cycle, collisions, downlink behavior, and battery cost.

6. Record the evidence state

Record spreading factor, bandwidth, coding settings, payload and preamble, conducted power, antenna path, RSSI, SNR, sensitivity source, airtime, retries, delivery, and ADR window.

7. Check yourself

Why is SF12 symbol time 32 times SF7?
Answer: Five SF steps double the state count five times, and 2^5 is 32.
Does 15.05 dB more ideal gain mean more transmit power?
Answer: No. It comes from longer spreading and receiver integration.
Does the Shannon value promise 1,794 bps?
Answer: No. It is a theoretical ceiling under the stated bandwidth and SNR.
Honesty boundary.

The bridge keeps ideal spreading arithmetic separate from radio and network evidence.

Computed
States, symbol time, coded-rate screen, ideal gain, slowdown, and capacity ceiling.
Specified
Bandwidth, coding fraction, spreading factor, SNR screen, radio, and packet settings.
Observed
Sensitivity, RSSI, SNR, airtime, retries, delivery, collisions, and battery use.

Correct, not complete: this page does not certify a radio, link, capacity plan, regulation, or deployment.

Return to LoRaWAN Introduction with margin and airtime kept together →