Lecture notes on turbo codes.

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The summations involved in the algorithm are calculated using trellis edges,rather than using pairs of states. This makes the algorithm general andcapable of dealing with parallel edges – suitable for TCM.The A-SISO algorithm at Bit LevelConsider a rate ½ convolutional encoder. U k input and C 1,k and C 2,k outputbits at time k, taking values {0,1}. Therefore on the trellis edges at time k wehave u k (e), c 1,k (e), c 2,k (e). Drop k for simplicity.Define the reliability (LLR) of a bit Z taking values {0,1} at time k asP [ Z = 1;.]λ [ Z;.]≡ logkk= πk[ Z = 1;.] − πk[ Z = 0;.]Pk[Z = 0;.]Eα k (s) = max (e:s (e)=s) {α k-1 [s S (e) + u(e)λ k [u;I] + c 1 (e)λ k [C 1 ;I] +c 2 (e)λ k [C 2 ;I] } + h αkβ k (s) = max (e:sS(e)=s) {β k+1 [s E (e) + u(e)λ k+1 [u;I] + c 1 (e)λ k+1 [C 1 ;I] +c 2 (e)λ k+1 [C 2 ;I] } + h βkwith initial values α 0 (s) = 0 if s=S 0 and α 0 (s) = - ∞ otherwise and β n (s)=0 ifs=S n and β n (s) = - ∞ otherwise. h αk and h βk are normalization constants.For the inner decoder, which is connected to the AWGN channel, we haveλ k [C 1 ;I]= (2A/σ 2 )r 1,k λ k [C 2 ;I]= (2A/σ 2 )r 2,k wherer i,k = A(2Z i - 1) +i ,k , i=1,2is the received samples at the output of the matched filter, c i ∈ {-1,1} andn i,k is the zero mean independent identically distributed (i.i.d.) Gaussiannoise samples with variance σ 2 .18
The extrinsic bit information for U,C 1 and C 2 can be obtained asλ k (U;O) = max (e:u(e)=1) {α k-1 [s S (e) + c 1 (e)λ k [C 1 ;I] + c 2 (e)λ k [C 2 ;I]+ β k [s E (e)]}- max (e:u(e)=0) {α k-1 [s S (e) + c 1 (e)λ k [C 1 ;I] + c 2 (e)λ k [C 2 ;I]+β k [s E (e)]}λ k (C 1 ;O) = max (e:c1(e)=1) {α k-1 [s S (e) + u(e)λ k [U;I] + c 2 (e)λ k [C 2 ;I]+ β k [s E (e)]}- max (e:c1(e)=0) {α k-1 [s S (e) + u(e)λ k [U;I] + c 2 (e)λ k [C 2 ;I]+β k [s E (e)]}λ k (C 2 ;O) = max (e:c2(e)=1) {α k-1 [s S (e) + u(e)λ k [U;I] + c 1 (e)λ k [C 1 ;I]+ β k [s E (e)]}- max (e:c2(e)=0) {α k-1 [s S (e) + u(e)λ k [U;I] + c 1 (e)λ k [C 1 ;I]+β k [s E (e)]}Parameters for PCCCIn general for a Convolutional code (CC) w min is defined as the minimuminformation weight in the error events of the CC.w min = 1 for non recursive codesw min = 2 for recursive codes.The error coefficient (in p b (error)) interleaving gain for a PCCC with largeinterleaving length goes as N 1-w min , where N is the interleaver length.19
Page 1 and 2: Concatenated Coding• Parallel Con
Page 3 and 4: This finishes the first or initial
Page 5 and 6: PCCC -Reference: A soft-Input Soft-
Page 7 and 8: Differences between PCCC and SCCC:P
Page 9: similarly βn(s) = 0 for s=S n othe

Lecture notes on turbo codes.

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