Lecture notes on turbo codes.

The relationship between these functions depends on the particular encoder.As an example, in the case of systematic encoders, (s E (e),c(e)) also identifiesthe edge since u(e) is uniquely determined by c(e).Here it is only assumed that the pair (s S (e), u(e)) uniquely identifies theending state s E (e) – this assumption is always verified , as it is equivalent tosay that – given the initial trellis state, there is a one-to-one correspondencebetween input sequences and state sequences.The Additive SISO Algorithm (A-SISO)αk⎡( s)= log⎢∑exp{α⎢ E⎣e: s ( e)= s⎤[ c(e);I]}⎥,k⎥⎦Sk −1 [ s ( e)]+ πk[ u(e);I]+ πk=1,2,... nβk⎡( s)= log⎢exp{⎢∑ βk+⎣e: sS( e)= sk = n −1,..,01[ sE( e)]+πk + 1[ u(e);I]+πk + 1⎤[ c(e);I]}⎥,⎥⎦These are forward and backward recursions. At time k, the output(extrinsic) probability distributions are computed as (approximately)πk⎡⎤⎢SE( c;O)= log ∑exp{αk−1 [ s ( e)]+ πk[ u(e);I]+ βk[s ( e)]}⎥⎢⎣e:c(e)= c⎥⎦⎡⎤⎢SEπk( u;O)= log ∑exp{αk−1 [ s ( e)]+ πk[ c(e);I]+ βk[s ( e)]}⎥⎢⎣e:u(e)= u⎥⎦with initial values α 0 (s) = 0 for s=S 0 otherwise α 0 (s) = -∞16