Design of unstructured and protograph-based LDPC coded continuous phase modulation

Abstract

In this paper, we derive an asymptotic analysis and optimization of coded CPM systems using both unstructured and protograph-based LDPC codes ensembles. First, we present a simple yet effective approach to design unstructured LDPC codes : by inserting partial interleavers between LDPC and CPM, and allowing degree-1 and degree-2 variable nodes in a controlled pattern, we show that designed codes perform that can operate very close to the maximum achievable rates. Finally, the extension to protograph based codes is discussed. We provide some simple rules to design good protograph codes with good threshold properties.

Protograph-based LDPC convolutional codes for Continuous phase modulation

“…For ease of presentation, we compare the codes with coupling factor L = 50 when concatenated with the CPM modulator Q CP M : quaternary, Gray mapping, L CP M = 2, Raised cosine pulse and h = 1/4. Concerning the optimized protograph codes, C (2) still have a very good performance [11]. The performance one can achieve when optimizing rate-1/2 unstructured LDPC codes for this CPM is 0.7dB [11], here we almost achieved this limit with C…”

“…We consider the following scheduling: an iteration ℓ consists in one BCJR forward backward recursion for the CPM soft decoder, followed by one BP data pass check pass for the LDPC-C code. We further assume partial interleavers between the CPM module and each VN bundle [11], [23].…”

“…otherwise, I ℓ c (q, r) = 0. Considering partial interleavers [11], at the end of each iteration, the a posteriori MI evaluated at the VN r is:…”

“…Taking advantage of this decomposition, CPM has greatly benefited from the concept of turbo-decoding. Several papers considered serial concatenations with convolutional codes [3]- [6] and Low Density Parity Check (LDPC) codes [7]- [11].…”

Precoding for Non-Coherent Detection of Continuous Phase Modulations

“…For the coherent case, it can be shown that EXIT curves of most widely spreaded CPM in use have EXIT charts reaching the point (1, 1), except if some particular precoding is used. When BICM CPM schemes are considered, this convergence enables the use of efficient turbo-detection schemes involving simple outer coding schemes based on convolutional codes such as in the DVB-RCS2 standard [8] or based on sparse graph based codes [11], [12]. For the non-coherent case, it has been pointed out by [9] that non-coherent EXIT curves do not converge to point (1, 1) contrary to the coherent case.…”