LoRa from low Earth orbit

Processing gain below the noise floor#

LoRa encodes $S\in \{7,\dots ,12\}$ bits per symbol as a linear frequency chirp swept across bandwidth $B\in \{125,250,500\}\text{kHz}$ over $2^S$ chips[1]. The symbol duration $T_s=2^S/B$ and receiver sensitivity $S_{\text{rx}}$ are:

$$T_s=\frac{2^S}{B},S_{\text{rx}}=-174+10{\log }_{10}(B)+\text{NF}+{\text{SNR}}_{\min }(S)[\text{dBm}]$$

where ${\text{SNR}}_{\min }$ steps from $-7.5\text{dB}$ at $\text{SF7}$ down to $-20.0\text{dB}$ at $\text{SF12}$[2]. Compared with non-coherent 2-FSK at $4800\text{Bd}$ (which requires $E_b/N_0\approx 13.4\text{dB}$ at $\text{BER}=10^{-5}$[3]), LoRa trades data rate for $10\text{-}18\text{dB}$ of link margin, allowing FossaSat-1 and NORBY to close UHF links into quarter-wave ground whips[4].

Mode	Req. SNR (dB)	Sensitivity (dBm)	Bit rate (bps)	32 B Time-on-Air (ms)	Static Doppler limit (Hz)
SF7 / 125 kHz	−7.5	−127.0	5,469	72	244
SF8 / 125 kHz	−10.0	−129.5	3,125	134	122
SF9 / 125 kHz	−12.5	−132.0	1,758	247	61
SF10 / 125 kHz	−15.0	−134.5	977	453	31
SF11 / 125 kHz	−17.5	−137.0	537	987	15
SF12 / 125 kHz	−20.0	−139.5	293	1,810	8

The zenith Doppler-rate limit#

A static frequency offset merely shifts the FFT peak by a constant number of bins, which the preamble resolver corrects up to $\pm 0.25B$. A time-varying Doppler shift $\dot{f}$ during a single chirp, however, smears energy across adjacent FFT bins of width $\Delta f_{\text{bin}}=B/2^S$[1]. During one symbol $T_s$, the frequency drifts by $\dot{f}{T}_s=\dot{f}{2}^S/B$. Requiring this drift to remain smaller than a fraction of one bin ($\Delta f_{\text{bin}}$) yields the Doppler-rate ceiling:

$$|\dot{f}{|}_{\max }\propto \frac{B^2}{2^{2S}}$$

At $525\text{km}$ on $436.7\text{MHz}$, an overhead pass reaches $|\dot{f}|\approx 160\text{Hz/s}$. Every step of $+1$ in spreading factor quarters the tolerable Doppler rate: $\text{SF7-SF10}$ at $125\text{kHz}$ pass cleanly through zenith, whereas $\text{SF12}$ at $125\text{kHz}$ ($T_s=32.8\text{ms}$) requires Low Data Rate Optimization (LDRO) or widening the bandwidth to $250\text{kHz}$[2].