Short note: Turbo coding

Turbo coding is a forward error-correction (FEC) technique, introduced by Berrou, Glavieux and Thitimajshima in 1993, that uses parallel concatenated recursive systematic convolutional (RSC) codes joined by an interleaver, with iterative soft decoding. It comes within about 0.5–1 dB of the Shannon limit.

Encoder

 d(k) --+------------------------------> x (systematic)
        +-->[RSC encoder 1]------------> y1 (parity 1)
        +-->[interleaver]->[RSC enc. 2]-> y2 (parity 2)
                 |
        [puncturing / MUX] -> rate 1/3 (or 1/2)

• The data bits are sent unchanged (systematic part).
• Encoder 1 produces parity from the data in natural order; encoder 2 produces parity from an interleaved (permuted) copy.
• The interleaver makes the two parity streams nearly independent, so a pattern that is weak for one encoder is strong for the other.
• Puncturing can raise the code rate from 1/3 to 1/2.

Decoder

     +--------------------------------------+
     v                                      |
 x,y1->[SISO decoder 1]->[interleave]->[SISO decoder 2]
                                  y2-->^        |
           extrinsic info  <-[de-interleave]<---+
                                   after N iterations
                                   -> hard decision

• Two soft-input soft-output decoders (MAP/BCJR or SOVA) exchange extrinsic information (log-likelihood ratios).
• Each iteration improves the reliability of the bit estimates; 4–8 iterations are typical.

Merits: near-Shannon-limit performance at low SNR. Demerits: decoding delay and complexity, and an "error floor" at very low BER. Used in 3G (WCDMA, CDMA2000), LTE data channels and deep-space links.