Log-Likelihood Ratio Calculation for Iterative Decoding on Rayleigh Fading Channels Using Padé Approximation

Table 3

Degree distribution profiles and their iterative thresholds for the uncorrelated flat Rayleigh fading channel with unknown CSI by the density evolution. These profiles are (a) threshold optimized and (b) rate optimized codes. The Shannon limit for rate half code is  𝜎=0.7436  (𝐸_𝑏/𝑁₀=2.572 dB).

(a) Threshold optimized

	True LLR	Taylor3	Pade23
𝜆2	0.200284000	0.200283000	0.200284000
𝜆3	0.228588000	0.228591000	0.228588000
𝜆7	0.067795000	0.067777920	0.067794487
𝜆8	0.210231000	0.210246680	0.210231613
𝜆30	0.292602000	0.292601400	0.292601850
𝜌9	1.000000000	1.000000000	1.000000000
Rate	0.500000	0.500000	0.500000
𝜎^⋆	0.7232423	0.7232385	0.7232422
𝐸𝑏/𝑁0^⋆ [dB]	2.7068537	2.7069429	2.7068549

(b) Rate optimized

	True LLR	Taylor3	Pade23
𝜆2	0.202860	0.202758	0.202860
𝜆3	0.214366	0.214369	0.214366
𝜆7	0.157257	0.157244	0.157258
𝜆8	0.112911	0.112922	0.112859
𝜆30	0.312606	0.312607	0.312607
𝜌9	1.000000	1.000000	1.000000
Rate	0.4946847	0.4946836	0.4946845
𝜎^⋆	0.740167	0.740167	0.740167
𝐸𝑏/𝑁0^⋆ [dB]	2.613406	2.613406	2.613406