Transmission of Bernoulli Sources Using Convolutional LDGM Codes

We propose in this paper to exploit convolutional low density generator matrix (LDGM) codes for transmission of Bernoulli sources over binary-input output-symmetric (BIOS) channels. To this end, we present a new framework to prove the coding theorems for linear codes, which unifies the channel coding theorem, the source coding theorem and the joint source-channel coding (JSCC) theorem. In the presented framework, the systematic bits and the corresponding parity-check bits play different roles. Precisely, the noisy systematic bits are used to limit the list size of typical codewords, while the noisy parity-check bits are used to select from the list the maximum likelihood codeword. This new framework for linear codes allows that the systematic bits and the parity-check bits are transmitted in different ways and over different channels. With this framework, we prove that the Bernoulli generator matrix codes (BGMCs) are capacity-achieving over BIOS channels, entropy-achieving for Bernoulli sources, and also system-capacity-achieving for JSCC applications. A lower bound on the bit-error rate (BER) is derived for linear codes, which can be used to predict the error floors and hence serves as a simple tool to design the JSCC system. Numerical results show that the convolutional LDGM codes perform well in the waterfall region and match well with the derived error floors, which can be lowered down if required by simply increasing the encoding memory.