Iterative Some Decoding algorithms for Linear Block codes
Niranjan Ratnakar December 14, 2002
1
Abstract
This report talks about Iterative decoding. It primarily addresses 2 scenarios, when Hard decision is done and the other case when additional soft information is available.
2
Introduction
The main idea behind this report is to show that the problem of decoding can be posed as problems, which have been well-researched, and can be solved using techniques that are well known. The reasonable assumptions made in order to pose the decoding problem as other problems are presented here. In this report, the solutions themselves are not given but references to papers which have the solutions are given. In the hard decision case, the decoding problem is posed as free energy minimisation. In the soft decision case, the problem is posed as a gradient descent algorithm.
3
Hard decision
The reference for this section is [1]. The decoding problem can be stated as follows. Consider three binary vectors: S of length N , and z and n of lengths M N , related by: (As + n)mod2 = z where A is a binary matrix. We can think of s as the information sequence. A is the generator matrix for the code. n is the noise added by the channel. z is the observed output at the channel. Our task would then be to infer s given z and A, and given assumptions about the statistical properties of s and n. A reasonable assumption that the probability distributions of s and n are
1
seperable is also made. That is P (s, n) =
n
P (sn )
m
P (nm )
It can be shown that the Maximum-A-Posteriori (MAP) rule tells us to choose the s that maximises P (s|z, A) = P (z|A)p(s) P (z|A) (1)
But this needs an exhaustive search. In order to simplify this problem, we try to approximate the quantity in Equation (1) by Q(s; ) = n qn (sn ; n ) such that 1 qn (sn = 1; n ) = 1 + e-n e-n qn (sn = 0; n ) = 1 + e-n The parameters are chosen so as to minimise the variational free energy F () =
s
Q(s; ) log
Q(s; ) P (z|s, A)P (s)
Choosing the parameters to minimise the variational free energy is a well known problem, which can be solved using iterative techniques. One way of solving this problem is outlined in [1]. Once the parameters are chosen, we see that the maximisation problem is solved trivially by choosing sn = 1 0 if n 0 , otherwise.
4
Soft Decision case
The reference for this section is [2]. In this section, we consider the case when soft information is available at the decoder. The channel is assumed an AWGN channel with single-sided noisepower spectral density N0 /2. The modulation used is BPSK modulation where 0 is mapped to 1 and 1 is mapped to -1. We have seen in class that log with Em (C ) = ln
bC bC (-1) n l=1,l=m bm
P (xm = +1|y) 4 = ym + Em (C ) P (xm = -1|y) N0 tanh
2yl No bl bl
(2)
(3)
n l=1,l=m
tanh
2yl No
2
The first term in Equation (2) is known as the intrinsic information and the second term is known as the extrinsic information. But the extrinsic information is difficult to evaluate if the rate of the code is not very large. To evaluate the extrinsic information, we make the following approximations: Since |tanh(.)| < 1, parity check vectors of low hamming weight have a larger contribution than vactors of large haming weight. Thus, the evaluation over parity check vectors of minimum weight would suffice. It is enough to compute the extrinsic information using the spanning vectors of C of minimum weight. The m-th position is affected only by parity check vectors having a 1 in the m-th position, represented as Bm . Let the extrinsic information be represented by Em (Bm ). Using these assumptions, the authors in [2], present the following algorithm Initialisation: Set Rm =
(0) 4 N0 ym , m
[1, n].
Set iteration counter i = 0, and define the maximum number of iterations . if [(stop criterion is not fulfilled) and (i < )] (*) for m = 1, . . . , n,compute:Em = for m = 1, . . . , n, update:
(i) bBm n l=1,l=m
tanh (yl ) ...