Turbo coding and 8×8 Hadamard code

Turbo coding

Turbo coding is a powerful forward error-correction scheme that uses two (or more) recursive systematic convolutional (RSC) encoders in parallel, separated by an interleaver, and is decoded iteratively by two soft-in soft-out decoders that exchange information. It performs within about 0.5–1 dB of the Shannon limit and is used in 3G (WCDMA, CDMA2000) and 4G LTE.

 d --+---------------------> systematic bits
     +-->[RSC encoder 1]---> parity 1
     +-->[interleaver]->[RSC encoder 2]-> parity 2

8 × 8 Hadamard code

A Hadamard (Walsh) matrix is built recursively (Sylvester construction), where H′ is the complement of H:

 H1 = [0]

 H2 = | H1  H1 |  = | 0 0 |
      | H1  H1'|    | 0 1 |

 H4 = | H2  H2 |  = | 0 0 0 0 |
      | H2  H2'|    | 0 1 0 1 |
                    | 0 0 1 1 |
                    | 0 1 1 0 |

 H8 = | H4  H4 |      (H' = complement of H)
      | H4  H4'|

Writing out H8:

         c1 c2 c3 c4 c5 c6 c7 c8
 W0:      0  0  0  0  0  0  0  0
 W1:      0  1  0  1  0  1  0  1
 W2:      0  0  1  1  0  0  1  1
 W3:      0  1  1  0  0  1  1  0
 W4:      0  0  0  0  1  1  1  1
 W5:      0  1  0  1  1  0  1  0
 W6:      0  0  1  1  1  1  0  0
 W7:      0  1  1  0  1  0  0  1

Each row is one 8-bit Walsh codeword. Any two rows differ in exactly 4 positions (in ±1 form, with 0 → +1 and 1 → −1, their inner product is 0), so the rows are orthogonal, as required for CDMA spreading codes.