Low Density Parity Check codes in OFDM system
Aravind R. Iyengar, Andrew Thangaraj, Srikrishna Bhashyam · 2009
In this work, we study the performance of Low Den- sity Parity Check (LDPC) codes over an Orthogonal Frequency Division Multiplexing (OFDM) channel. We state a concentra- tion theorem which shows that no Gaussian approximation is required in the analysis of LDPC codes over OFDM. Then we propose a rigorous density evolution method (without Gaussian approximations) to prove the existence of thresholds for LDPC codes over OFDM and evaluate the thresholds for various regular and irregular LDPC codes. We calculate the capacity of OFDM channel and compare LDPC threshold with this theoretical limit and show that for irregular codes, LDPC thresholds are very close to capacity at higher rates. We also compare the LDPC threshold in OFDM with LDPC threshold in an ISI channel with BCJR equalization. Using the feedback to the transmitter, we apply Mercury/Waterfilling power allocation to improve the OFDM capacity and LDPC thresholds. We show that with Mercury/Waterfilling power allocation LDPC thresholds are very close to capacity even at moderate rates . I. INTRODUCTION subcarriers, (3) More accurate computation of thresholds for ir- regular and regular LDPC codes, (4) Comparison with OFDM, ISI capacities. Specifically, we state a concentration theorem which shows that no Gaussian assumption is necessary in the analysis of LDPC codes over OFDM. Using this result we then propose a rigorous density evolution algorithm to compute threshold for LDPC codes over an ISI channel under OFDM. We assume that one code block is transmitted using a single OFDM symbol. In the algorithm, we allow the block length to tend to infinity. Consequently, the subcarrier spacing reduces and the number of subcarriers tend to infinity for the same bandwidth. Since the number of subcarriers tend to infinity, the finite cyclic prefix results in no additional overhead. We calculate the OFDM channel capacity and compare OFDM thresholds obtained by our density evolution with this theoretical limit. We show that for higher rates (rates higher than 0.6) the thresholds are very close to the theoretical limit. An optimum power allocation scheme, Mercury/Waterfilling, for parallel Gaussian channel with arbitrary input constellation has been proposed by Lozano et al (10). We use this power allocation scheme to improve the OFDM capacity. We apply LDPC codes with this power allocation and demonstrate that LDPC thresholds also show considerable improvement. We show that, with this optimum power allocation, LDPC thresholds are very close to capacity even at moderate rates ( rates higher than 0.2). We also make a comparison between the time-domain BCJR algorithm and the frequency-domain OFDM method for equalizing an ISI channel. To this end, we compare the threshold for LDPC codes under OFDM with that of the BCJR thresholds obtained by using the algorithm given in (3).