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 · 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.

Symbol-Time Arithmetic

1. Symbol-time arithmetic. The chapter uses a 125 kHz LoRa bandwidth and the shortcut T_symbol = 2^SF / 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 R_s = bandwidth / 2^SF. A simple bit-rate intuition is R_b = SF x bandwidth / 2^SF. 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.