# Soft Iterative Decoding Algorithms for Rateless Codes in Satellite Systems
## 4 Simulation Results  
We first investigate the performance of the cross-layer coding scheme. As a reference performance, the bit error rate (BER) performance with the tandem soft iterative decoding is first estimated. In order to investigate the performance of the proposed HARQ for the incremental redundancy with FEC scheme, we employ a simple binary phase shift keying (BPSK) modulation. In the MAC layer, the LT codes with a degree distribution specified in [17] were used, while in the PHY layer the LDPC code specified in DVB-S2 with length of \(N=16200\) and code rate of \(1/4\) is applied. The maximum number of iterations for both the MAC-layer LT code and the PHY-layer LDPC codes are set to 40. We assume \(k\leq K\) to ensure the cross-layer decoding attempt once the decoding is finished by the first LDPC decoder.  
Figure 8 shows the BER performance comparisons of the proposed cross-layer joint iterative decoding scheme with \(\eta_{max}=2\) compared to the conventional tandem decoding and non-systematic LT codes, when \(k=K\) and \(k=K/2\). We first compare the performance over an additive white Gaussian noise (AWGN) channel, for relative performance comparison of decoding schemes applied to the cross-layer codes. In the figure, \(R^{-1}=k/n\), denotes the inverse of the code rate of the LT code. As shown in the figure, regardless of \(R^{-1}\), the proposed decoding method outperforms the existing tandem decoding, which achieves better performance than the case with a non-systematic code. Then, a satellite rain fading channel with a Ka frequency band is applied [20]. Dynamic rain fading data with a second interval is generated, and the maximum fading value is set to 28 dB. The simulation results over a rain fading channel in Figure 8 demonstrate exactly the same trend we found in the AWGN channel, so the proposed decoding method for the cross-layer code outperforms the others.  
Figure 7: Flow chart of the joint soft iterative decoding algorithm for cross-layer rateless codes.  
Next we assess the performance of the proposed rateless LDPC coding scheme over the satellite rain fading channel. The BER performance of the proposed HARQ scheme is compared with that of the conventional LDPC codes, where the proposed HARQ accomplishes type-II incremental redundancy with rateless LDPC codes. Figure 9 compares the BER performance of the proposed scheme over the rain fading satellite channel with a Ka frequency band [20]. In the simulation, short frame sized LDPC codes defined in the DVB-S2 are used with a number of code rates, \(R\), and rain attenuation data of about 12 hours are synthesized with a time interval of second unit and with a maximum fading value of 24 dB. The maximum numbers of iterations are limited to 20, 40, and 3 for LT decoding, LDPC decoding, and joint iterative decoding, respectively. The maximum number of retransmissions allowed is determined by when the inverse of the code rate of the LT code, \(R^{-1}\), reaches up to 4.  
From the simulation, the average \(R^{-1}\) values observed from all the rateless codes range from 1.000072 to 1.014061, implying that additional redundancies used can almost be ignored compared to the conventional LDPC codes. Considering this, the results in Figure 9 demonstrate that, with the use of rateless codes, rain fading can be effectively compensated for without an appreciable amount of additional resources being required. For example, the rateless LDPC code of which the component LDPC code has an \(R\) of 7/9 can produce a better performance than the conventional LDPC code with an \(R\) of 4/9, and produces an almost identical performance to that of the conventional LDPC code with an \(R\) of 1/3.  
Figure 8: Bit error rate (BER) performance of the proposed cross-layer rateless codes over an additive white Gaussian noise (AWGN) and rain fading channel.