Editing Turbo code (section)

==The decoder==
The decoder is built in a similar way to the above encoder. Two elementary decoders are interconnected to each other, but in series, not in parallel. The <math>\textstyle DEC_1</math> decoder operates on lower speed (i.e., <math>\textstyle R_1</math>), thus, it is intended for the <math>\textstyle C_1</math> encoder, and <math>\textstyle DEC_2</math> is for <math>\textstyle C_2</math> correspondingly. <math>\textstyle DEC_1</math> yields a [[Turbo code#Soft decision approach|soft decision]] which causes <math>\textstyle L_1</math> delay. The same delay is caused by the delay line in the encoder. The <math>\textstyle DEC_2</math>'s operation causes <math>\textstyle L_2</math> delay.

[[File:turbo decoder.svg]]

An interleaver installed between the two decoders is used here to scatter error bursts coming from <math>\textstyle DEC_1</math> output. ''DI'' block is a demultiplexing and insertion module. It works as a switch, redirecting input bits to <math>\textstyle DEC_1</math> at one moment and to <math>\textstyle DEC_2</math> at another. In OFF state, it feeds both <math>\textstyle y_{1k}</math> and <math>\textstyle y_{2k}</math> inputs with padding bits (zeros).

Consider a memoryless [[additive white Gaussian noise|AWGN]] channel, and assume that at ''k''-th iteration, the decoder receives a pair of random variables:

:<math>\begin{align}
  ~x_k &=  (2d_k - 1) + a_k\\
  ~y_k &= 2( Y_k - 1) + b_k
\end{align}</math>

where <math>\textstyle a_k</math> and <math>\textstyle b_k</math> are independent noise components having the same variance <math>\textstyle \sigma^2</math>. <math>\textstyle Y_k</math> is a ''k''-th bit from <math>\textstyle y_k</math> encoder output.

Redundant information is demultiplexed and sent through ''DI'' to <math>\textstyle DEC_1</math> (when <math>\textstyle y_k = y_{1k}</math>) and to <math>\textstyle DEC_2</math> (when <math>\textstyle y_k = y_{2k}</math>).

<math>\textstyle DEC_1</math> yields a soft decision; i.e.:

:<math>\Lambda(d_k) = \log\frac{p(d_k = 1)}{p(d_k = 0)}</math>

and delivers it to <math>\textstyle DEC_2</math>. <math>\textstyle \Lambda(d_k)</math> is called the ''logarithm of the likelihood ratio'' (LLR). <math>\textstyle p(d_k = i),\, i \in \{0, 1\}</math> is the ''a posteriori probability'' (APP) of the <math>\textstyle d_k</math> data bit which shows the probability of interpreting a received <math>\textstyle d_k</math> bit as <math>\textstyle i</math>. Taking the ''LLR'' into account, <math>\textstyle DEC_2</math> yields a hard decision; i.e., a decoded bit.

It is known that the [[Viterbi algorithm]] is unable to calculate APP, thus it cannot be used in <math>\textstyle DEC_1</math>. Instead of that, a modified [[BCJR algorithm]] is used. For <math>\textstyle DEC_2</math>, the [[Viterbi algorithm]] is an appropriate one.

However, the depicted structure is not an optimal one, because <math>\textstyle DEC_1</math> uses only a proper fraction of the available redundant information. In order to improve the structure, a feedback loop is used (see the dotted line on the figure).