# Update rule in Soft Output Viterbi Algorithm (SOVA) -Hagenauer rule

Soft output of SOVA, suggested by Hagenauer http://www.cse.lehigh.edu/~jingli/teach/S2003turbo/notes/SOVA.pdf , says in 2.2.2 "The Soft-Output Viterbi Algorithm" that

$$L_{j}^{\verb|^|}:= f(L_{j}^{\verb|^|}$$, $$\Delta$$), if $$u_{k}\neq u_{k-U}$$ for $$U = 0,1,..,\delta$$ where $$\delta$$ is the traceback depth.

It is also given in literature that, https://www.eit.lth.se/sprapport.php?uid=362,equation 1.26, 1.27, page 11, the Hagenauer rule can be given by

$$L(u_{k}^{\verb|^|}) \approx u_{k}^{\verb|^|}min(\Delta_{k+l})$$ where l = 0,1,..,U

This https://patentimages.storage.googleapis.com/ea/4a/6a/c8f50b500f6f42/WO2011020770A1.pdf says in the background section that "The update rule applied for the reliability values affects the quality (e.g., accuracy) of the extrinsic soft output values and thus the overall performance of the iterative decoder."

1. I'm unable to find mathematical equivalence between the two given equations. I've gone through Log likelihood Algebra (e.g., given by Todd K. Moon - Error Correction Coding- Mathematical Methods and Algorithms 2005 and others available). How can I get one from the other mathematically?
2. It says in literature that choice of the two given equations depends on applications. I can understand that the second equation gives very optimistic values of LLRs and hence accuracy is compromised. Can you give me any example where one is preferred over the other?

Any hint/suggestion/literature is appreciated.