Design and Analysis of LDGM-Based Codes for MSE Quantization
  Approaching the 1.5329-dB shaping (granular) gain limit in mean-squared error (MSE) quantization of R^n is important in a number of problems, notably dirty-paper coding. For this purpose, we start with a binary low-density generator-matrix (LDGM) code, and construct the quantization codebook by periodically repeating its set of binary codewords, or them mapped to m-ary ones with Gray mapping. The quantization algorithm is based on belief propagation, and it uses a decimation procedure to do the guessing necessary for convergence. Using the results of a true typical decimator (TTD) as reference, it is shown that the asymptotic performance of the proposed quantizer can be characterized by certain monotonicity conditions on the code's fixed point properties, which can be analyzed with density evolution, and degree distribution optimization can be carried out accordingly. When the number of iterations is finite, the resulting loss is made amenable to analysis through the introduction of a recovery algorithm from ``bad'' guesses, and the results of such analysis enable further optimization of the pace of decimation and the degree distribution. Simulation results show that the proposed LDGM-based quantizer can achieve a shaping gain of 1.4906 dB, or 0.0423 dB from the limit, and significantly outperforms trellis-coded quantization (TCQ) at a similar computational complexity. 
Qingchuan Wang, Chen He