Comparison of belief propagation and iterative threshold decoding based on dynamical systems

Abstract

For a special class of convolutional codes, iterative threshold decoding (ITD) has been shown by simulations to achieve the same error rate performance as belief propagation (BP). In order to get a better understanding of these iterative decoding algorithms, we describe ITD and BP as discrete-time dynamical systems. Based on the theory of dynamical systems, we compare the dynamical behavior of ITD and BP. For the special case of a linear dynamical system, the behavior can be completely characterized. In this case we show that the fixed points of both ITD and BP are globally stable but they do not coincide. The analysis is extended to the case of a continuous-time dynamical system, which represents an important step for modeling analog iterative decoders.

A Continuous-Time Recurrent Neural Network for Joint Equalization and Decoding – Analog Hardware Implementation Aspects

“…The dynamical behavior of belief propagation can be described based on Eqs. (16), (18) and Figures 14,15, and 16 [45]…”

“…The absolute stability of belief propagation Eqs. (20), (21) was proven for repetition codes (one of the simplest binary linear block codes) in [44,45]. In this case…”

“…(17) where n =7,k =4.Thebelief propagation algorithm iteratively exchanges "messages" between parity and check nodes. For every binary linear block code characterized by the binary parity check matrix H of size (n − k) + n for n > k, three binary matrices P n h • n h , S n h • n h and B n h • n can be uniquely defined [44,45] such that Eq. (16) and Figure 15 perform continuous-time belief propagation if the following relations are fulfilled: A Continuous-Time Recurrent Neural Network for Joint Equalization and Decoding -Analog Hardware... http://dx.doi.org/10.5772/63387 u ¼ L,…”

“…This was motivated from a circuit design aspect, where ϒ d (the same is valid for ϒ e ) can be seen as a bandwidth limitation of the analog circuit, realized taking advantage of the low-pass filter behavior of transmission lines Figure 17. We have shown in [45] that the model in Figure 17 also has important dynamical properties when compared with the discrete-time belief propagation Eq. (21) [44].…”