Near-Capacity Variable-Length Coding: Regular and EXIT-Chart-Aided Irregular Designs

Appendix C

APP Channel Equalization

The APP equalizer outputs the channel information, which is expressed as {L_f(d_k)}, associated with all data bits, given N_c received samples hosted by the vector y = y₁^N_c as well as the a priori information {L_b(d_k)} corresponding to all data bits. The feed-forward information L_f(d_k) corresponding to bit d_k is given by

(C.1)

where L(d_k|y) =ln[P(d_k = + 1|y)/P(d_k = −1|y)] represents the a posteriori Logarithmic Likelihood Ratio (LLR) of the data bit d_k, which may be computed using the algorithm described below by following the approaches outlined in [69, 70, 72, 99].

The a posteriori LLR of thedatabit d_k can be expressed in conjunction with the channel states s_k−1 and s_k as

(C.2)

The sequence y1Nc involved in the computation of the probability P(sk−1, sk, y1Nc) can be written in a more detailed form as P(sk−1, sk, y1k−1, yk, yNck+1). Applying the chain rule for joint probabilities, i.e., exploiting that we have P(a, b) = P(a)P(b|a), this expression yields the following decomposition of P(sk−1, sk, y1Nc):

(C.3)

Let us introduce the following definitions:

(C.4)

(C.5)

(C.6)

Then according to [48] the term α_k(s) can be computed via the recursion

(C.7)

associated ...