# Turbo decoding algorithms Algorithms for Iterative (Turbo) Data ...

Turbo decoding algorithms Algorithms for Iterative (Turbo) Data ...

Kalle Ruttik 2005Kalle Ruttik 2005S-72.630 Algorithms for Turbo Decoding (5) 25S-72.630 Algorithms for Turbo Decoding (5) 26Modification of MAP algorithm· MAP algorithm operates in probability domain.p A k (u) = 1p(Y N1 )∑e:u(e)=uA k−1(sSk(e) ) · M k (e) · B k(sEk(e) )· When probablity is expressed by loglikelihood value wehave to deal with numbers in very large range. (overflowsin computers).Simplification Log-MAP algorithm description• Log-MAP algorithm is a transformation of MAP intologarithmic deomain.• The MAP algorithm logarithmic domain is expressed withreplaced computations- Multiplication is converted to addition.- Addition is converted to a max ∗ (·) operation.max ∗ (x, y) = log (e x + e y ) = max (x, y) + log(1 + e −|x−y|)• The terms for calculating the probabilities in trellis areconvertedα k (s) = log A k (s)β k (s) = log B k (s)γ k (s) = log M k (e)Kalle Ruttik 2005S-72.630 Algorithms for Turbo Decoding (5) 27Kalle Ruttik 2005S-72.630 Algorithms for Turbo Decoding (5) 28• The complete logMAP∑algortihm isλ A (k (u) = log A k−1 sSk (e) ) (· M k (e) · B k sEk (e) )− log∑e:u(e)=0= max ∗e:u(e)=1e:u(e)=1A k−1(sSk (e) ) · M k (e) · B k(sEk (e) )(αk−1(sSk (e) ) + γ k (e) + β k(sEk (e) ))( (− max ∗ αk−1 sSk (e) ) (+ γ k (e) + β k sEk (e) ))e:u(e)=0∑ (α k (s) = log A k sSk(e) ) · M k (e)e:s E k∑(e)=s (β k (s) = log B k+1 sSk+1(e) ) · M k+1 (e)e:s S k+1 (e)=sMax-Log-MAP decoding Algorithm• In summation of probabilities in Log-MAP algorithm weare using max ∗ (·) operation.• The max ∗ (·) requires to convert LLR value intoexponential and after adding 1 to move back into logdomain.• Simplifications- We can replace log ( 1 + e −|x−y|) by a lookup table.- We can skip the term Max-Log-Map.

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