LoRa chirp spread spectrum

Each LoRa symbol is a chirp: a tone that sweeps across the whole bandwidth in 2^SF chips. The data is carried by where the sweep starts; when it reaches the top it wraps to the bottom. A packet opens with a run of plain up-chirps (the preamble), two sync-word symbols, two and a quarter down-chirps, then the header and payload. Well below 0 dB SNR the sweeps disappear into the noise in this view, yet the receiver can still decode them.

Before modulation the bytes are whitened (XORed with a fixed pseudo-random sequence so long runs of equal bits do not repeat), a CRC-16 is appended, and a header carries the length and coding rate with its own 5-bit checksum. Each 4-bit nibble becomes a Hamming codeword of 5 to 8 bits. The interleaver then spreads every codeword diagonally across several symbols, so one badly received symbol damages only one bit of each codeword. Last, Gray mapping makes a one-bin error in the receiver flip only one bit.

The receiver does not know when a packet starts. It cuts the signal into symbol-long windows, dechirps each one and takes its FFT. Inside the preamble every window holds the same chirp, so the FFT peak lands in the same bin again and again, whatever the timing. A run of equal peaks means a packet. The bin of that peak mixes timing error and carrier offset; the down-chirps shift the other way under a timing error but the same way under a frequency offset, so up- and down-chirp peaks together separate the two. The phase step of the peak from one preamble symbol to the next gives the fraction of a bin.

Multiplying each symbol by the complex conjugate of a plain up-chirp cancels the sweep. What is left is a steady tone whose frequency is the symbol's value: the chirp's starting point becomes a frequency. Down-chirps are multiplied by an up-chirp instead. This view uses the receiver's own timing and frequency estimates from stage 3, so every symbol turns into a flat line.

An FFT of the dechirped symbol gathers all 2^SF chips into one bin. The noise stays spread across all bins, so measured against the noise in one bin the peak rises by the SNR plus 10·log₁₀(2^SF) dB of processing gain: 21 dB at SF7, 36 dB at SF12. That processing gain is how LoRa decodes below the noise floor. The receiver simply takes the largest bin as the symbol value.

The receiver runs the transmit chain backwards: symbol values, Gray code, de-interleaving, Hamming decoding, de-whitening. The first eight symbols always use the most robust settings (coding rate 4/8, two fewer bits per symbol) so the header gets through; the header then says how long the payload is and which coding rate it uses. Rates 4/7 and 4/8 correct one bit per codeword; 4/5 and 4/6 can only detect errors. The CRC-16 decides whether the packet is accepted.

Each step up in spreading factor doubles the symbol length and buys about 2.5 to 3 dB of margin, at roughly half the bit rate. The sweep sends random symbols through noise at every SNR from −30 to +5 dB for SF7 to SF12 and counts wrong FFT peaks, with perfect timing. The solid lines are the theory for picking the largest of 2^SF bins in white noise. The triangles mark the demodulator SNR limits listed in Semtech's SX1276 datasheet; a real radio also has to find and synchronize each packet, which the sweep leaves out.

LoRa devices transmit whenever they like, so packets overlap. When two packets use the same spreading factor, each receiver window can hold two FFT peaks and the receiver keeps the larger one: the stronger packet usually survives (the capture effect) and the weaker is lost; at similar power both can fail. Packets with different spreading factors barely interfere, because a chirp with the wrong slope does not collapse into one bin; it spreads over all of them. This receiver listens on every spreading factor in use, like a gateway.