The Spreading-Factor Numbers: Symbol Time, Coded Rate, and Airtime Pressure

The Spreading-Factor Numbers: Symbol Time, Coded Rate, and Airtime Pressure

Ada re-derives the chapter’s symbol times, symbol and bit rates, and the airtime pressure a higher spreading factor creates

foundations
math-foundations
lorawan
spreading-factor
intermediate
Ada ADA · CALCULATION AUDIT

The Spreading-Factor Numbers: Symbol Time, Coded Rate, and Airtime Pressure

Spreading factor is a physics tradeoff: more chirp states can help weak-link decoding, but each symbol occupies the shared channel for longer.

Turn one radio knob — the spreading factor — and two things move at once: a weaker signal may become readable, but each symbol holds the shared channel longer. At a 125 kHz bandwidth the chapter clocks an SF7 symbol at 1.024 ms and an SF12 symbol at 32.768 ms, about 32 times longer, so if 200 sensors all default to the slowest setting the shared channel, not one link, becomes the problem. This audit re-derives those symbol times, rates, and the airtime pressure a higher spreading factor creates.

Companion to the chapter LoRa Modulation and Spreading Factors — every number here comes from that chapter.

See the relationship before changing it

The figure reads from left to right. The blue card is lora spreading factor. The middle card applies this page's rule. The green card is symbol duration. 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 lora spreading factor, so the numeric fixture does not switch without explanation.

LoRa spreading factor changes symbol duration An input card leads through the rule symbol time = 2^SF / 125,000 Hz x 1,000 to the symbol duration result. INPUT PAGE INPUT APPLY THE RULE predict calculate check units OUTPUT RESULT
Walk the arrows. Each spreading-factor step doubles symbol time when bandwidth stays fixed.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline is 12 SF.

  2. 2

    Name the relationship. symbol time = 2^SF / 125,000 Hz x 1,000

  3. 3

    Substitute with units. 2^12 / 125,000 x 1,000 = 32.768 ms

  4. 4

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

Predict, then change lora spreading factor

Try Predict the direction of symbol time = 2^SF / 125,000 Hz x 1,000. Test another lora spreading factor, then compare symbol duration.

12 SF
Chapter baseline
Symbol duration

Observe Each spreading-factor step doubles symbol time when bandwidth stays fixed. Reset lora spreading factor to 12 and compare symbol duration.

Explain Each spreading-factor step doubles symbol time when bandwidth stays 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 lora spreading factor moves here. Field effects named in the technical boundary stay fixed.

Honesty note. This teaching model changes only lora spreading factor. It is not a field prediction.

TryAt Calculate, compare SF7 and SF12 with Bandwidth fixed at 125 kHz.
ObserveSymbol time expands from 1.024 to 32.768 ms and the coded bit-rate readout falls to only a few hundred bit/s.
ExplainBecause symbol duration is 2 raised to SF divided by bandwidth, each added spreading-factor step doubles airtime while improving demodulation sensitivity.

Ready: use the stated baseline inputs, then compare each displayed result.

Symbol-Time Arithmetic

1. Symbol-time arithmetic. The chapter uses a 125 kHz LoRa bandwidth and the shortcut Tsymbol = 2SF / bandwidth. Convert seconds to milliseconds by multiplying by 1000.

SF7: 2^7 / 125,000 = 128 / 125,000 = 0.001024 s = 1.024 ms
SF12: 2^12 / 125,000 = 4096 / 125,000 = 0.032768 s = 32.768 ms

Airtime-Pressure Check

2. Airtime-pressure check. Before payload, preamble, headers, coding-rate overhead, acknowledgements, or retries, one SF12 symbol lasts 32.768 / 1.024 = 32 times as long as one SF7 symbol. Compared with SF10, 32.768 / 8.192 = 4, so the chapter's fourfold SF10-to-SF12 warning is a direct consequence of the same equation.

Rate Arithmetic

3. Rate arithmetic. Symbol rate is Rs = bandwidth / 2SF. A simple bit-rate intuition is Rb = SF x bandwidth / 2SF. Applying the common introductory 4/5 coding-rate screen shows why SF7 is near 5.5 kbit/s and SF12 is only a few hundred bit/s before LoRaWAN packet overhead.

Check Arithmetic Result
SF7 symbol rate 125,000 / 128 976.56 symbols/s
SF12 symbol rate 125,000 / 4096 30.52 symbols/s
SF7 raw bit-rate intuition 7 x 125,000 / 128 6835.94 bit/s
SF7 with 4/5 coding-rate screen 6835.94 x 4 / 5 5468.75 bit/s, about 5.5 kbit/s
SF12 raw bit-rate intuition 12 x 125,000 / 4096 366.21 bit/s
SF12 with 4/5 coding-rate screen 366.21 x 4 / 5 292.97 bit/s, a few hundred bit/s

Release meaning: the arithmetic does not approve or reject a spreading factor by itself. It tells the review which evidence must be present: payload size, cadence, retry behavior, regional duty limits, ADR ownership, and a retest trigger if placement or gateway conditions change.

Every number above is taken from the chapter’s own spreading-factor example and re-derived step by step.

Technical boundaries. Chirp waveforms, packet headers, preambles, coding errors, collisions, capture, acknowledgements, and retries are not simulated by this symbol-rate arithmetic.