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

Table 1

The coefficients of LLR approximation functions for 8-PAM at SNR = 7.91 (𝜎=1.699).

(a) Taylor series of order 𝑛=3

	𝑦0	𝑐0	𝑐1	𝑐2	𝑐3
bit 1	0.0	0.0	1.2135	0.0	0.0241
bit 2	3.3449	2.3273	−0.7706	0.0205	0.0006
bit 3 (a)	1.8848	−0.7855	0.0544	0.2848	−0.0491
bit 3 (b)	6.9832	2.2528	−0.2497	−0.0181	0.0011

(b) Padé approximation of numerator

	𝑦0	𝑎0	𝑎1	𝑎2	𝑎3	𝑎4	𝑎5	𝑎6	𝑎7
bit 1	0.0	0.0	1.2135	0.0	0.0334	0.0	0.0025	0.0	2.61𝐸−05
bit 2	3.3449	3.3382	−1.5496	0.1836	0.0039	−0.0028	—	—	—
bit 3 (a)	1.8848	−1.1396	0.6046	—	—	—	—	—	—
bit 3 (b)	6.9832	5.6135	−3.1331	0.5963	−0.0481	0.0017	—	—	—

(c) Padé approximation of denominator

	𝑦0	𝑏0	𝑏1	𝑏2	𝑏3	𝑏4
bit 1	0.0000	1.0000	0.0000	0.0076	0.0000	0.0016
bit 2	3.3449	1.6585	−0.3831	0.0557	—	—
bit 3 (a)	1.8848	1.4533	−0.6204	0.3220	−0.0852	0.0113
bit 3 (b)	6.9832	1.3191	−0.5502	0.0496	−0.0026	—