LDPC Codes for Binary Asymmetric Channels Ninoslav Marina Department of Electrical Engineering University of Hawai‘i at M¯ anoa 2540 Dole Street, Honolulu, HI 96822 Abstract In this paper we examine the use of low-density parity- check (LDPC) codes in binary asymmetric channels. The problem is interesting since in some types of binary in- put fading multiple access channels (MAC) without chan- nel state information (CSI), which use the successive de- coding scheme, some of the users may experience asymmet- ric channels. It is well known that the successive decoder is a set of single user decoders and since there is no CSI at the receiver, the corresponding single user channels may be asymmetric in general. In that case the rate of inter- est of the user with the asymmetric channel is achieved by an unbalanced input distribution of the binary input. We are interested in constructing LDPC codes for these chan- nels that approach the desired rate tuple in the capacity re- gion. A convenient way of making the output distribution unbalanced is by introducing a mapper at the output of the encoder. Here we explain the method of mapping and its effect on the iterative decoding and derive closed form ex- pressions forthe upper bound of the probability of error. 1. Problem Statement Consider the binary multiple access channel Y = X 1 ⊕ A 2 X 2 , studied in [1], where Y,A 2 ,X 1 ,X 2 ∈{0, 1}. Its capacity region is given by R 1 (p 2 ) = 1 - h(αp 2 ) R 2 (p 2 ) = h(αp 2 ) - p 2 h(α) (1) for p 2 ∈ [0,p ∗ 2 ], where α = Pr{A 2 =1}, p 2 = Pr{X 2 = 1} and p ∗ 2 =  α  1+2 h(α)/α  -1 . It is described in Fig. 1. This channel can be decomposed into two single user channels. The channel between user 1 and the output, con- sidering user 2 as noise, is a binary symmetric channel with This work was initially presented in [1] and was partially supported by the Swiss National Science Foundation Grant #21-055699.98. 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 R 1 R 2 1-h(α p 2 * ) h(α p 2 * )-p 2 * h(α) A B Figure 1. Capacity region of the channel Y = X 1 ⊕ A 2 X 2 for α =0.615. crossover probability αp 2 . The channel between user 2 and the output, given that user 1 is decoded, is a Z -channel with p(0|1) = 1 - α, which is an asymmetric channel. Constructing a code for the BSC is well studied in litera- ture. Here we are interested in designing LDPC codes for the asymmetric binary channels like the Z -channel. For an asymmetric channel we have to use unbalanced codes, codes in which Pr{X =1} < 1/2. How can we use a linear LDPC code to obtain an unbalanced code that ap- proaches a desired rate of an asymmetric channel? Here we consider the Z -channel, but the analysis can be extended to other asymmetric channels. The capacity of the Z -channel with p(1|1) = α is C Z (α)= h  1 1+2 h(α)/α  - h(α) α  1+2 h(α)/α  , 978-1-4244-2036-0/08/$25.00 (c)2008 IEEE. Authorized licensed use limited to: UNIVERSITY OF OSLO. Downloaded on May 20, 2009 at 12:50 from IEEE Xplore. Restrictions apply.