Field Guide · algorithm

Also known as: LoRa interleaver, diagonal interleaver, offset interleaver

The LoRa diagonal interleaver is the bit-permutation that spreads each forward-error-correction codeword across all the symbols of a block along a rotating offset, so a channel burst that wipes out one whole symbol damages only a single bit of each codeword — exactly the pattern the Hamming code around it can repair.1 It is one link in the LoRa physical-layer chain: Gray mapping, then this interleaving, then Hamming(4+CR, 4) coding.2

symbols (columns) → codewords ↓ symbol[k] bit ((r+k) mod PPM) = codeword[r] bit k each codeword walks a diagonal
Bit k of codeword r lands in symbol k at bit position (r+k) mod PPM; successive codewords are staggered by one position, so their bits trace diagonals and no symbol holds two bits of the same codeword.

How it works

A LoRa interleaver block is a rectangle: PPM rows (the spreading factor — or SF−2 in a reduced-rate block) and (4+CR) columns, where CR is the coding rate 1..4. GopherTrunk’s Interleave and Deinterleave are exact inverses built on one rule:

symbol[k] bit ((r+k) mod ppm) = codeword[r] bit k

That +k offset per column is what makes it diagonal rather than a plain transpose: as the column index k advances, the bit is rotated one more position around the symbol, so a single codeword’s bits are smeared across every symbol at staggered positions. The payoff is fading resilience — corrupting one received symbol removes at most one bit from any codeword, leaving the per-codeword Hamming decoder a single-error problem it can solve.

Around the interleaver sit two more transforms, all inverse pairs so the whole chain is round-trip testable:

Stage TX direction RX direction
Gray mapping v ^ (v>>1) cumulative XOR
Diagonal interleave Interleave Deinterleave
FEC HammingEncode4 HammingDecode4

The Gray code mapping ensures adjacent chirp frequencies differ by one bit, so a small frequency slip in demodulation costs a single bit rather than several.

Coding rates and the Hamming code

The FEC is a family of Hamming(4+CR, 4) codes selected by the coding-rate field, carrying four data bits per codeword and CR parity bits:

CR Rate Parity Capability
1 4/5 1 (even parity) detect only
2 4/6 2 detect
3 4/7 3 (Hamming(7,4)) single-error correct
4 4/8 4 (Hamming(7,4) + overall) correct + double-error detect

At CR 3 and 4, GopherTrunk computes a 3-bit syndrome from the parity equations and maps it to the flipped bit position to correct it; CR 4 adds an overall-parity bit so a corrected single error can be told apart from an uncorrectable double error. The explicit header is always sent at the most robust rate, 4/8, because a receiver must decode it — to learn the payload length, coding rate and whether a payload CRC follows — before it knows the payload’s own coding rate.

Calibration caveat

As with the rest of LoRa’s reverse-engineered PHY, GopherTrunk implements these transforms as a self-consistent, round-trip-verified chain and flags exact bit-level interoperability with Semtech silicon — the interleaver’s rotation sign and the codeword bit order — as a calibration step gated behind captured golden vectors, not something assumed correct from the datasheet.

Sources

  1. LoRa — Wikipedia, on the LoRa CSS physical layer, its interleaving and coding rates. ↩

  2. Hamming code — Wikipedia, on the single-error-correcting block code the LoRa FEC is built from. ↩

See also