At the end of this chapter, we compare the effect of different BCH codes concatenated with the same QC(2286,1914) LDPC code (respectively, QC(4590,3835) LDPC code) by simulations in comparison with the predictions of our approximation formula. The results are summarized in Figures 4.8–4.11. The behaviors of the curves in these figures are seemingly synchronized. They all show that a larger BCH code is favored from the aspect of improving the error floor.
2 2.5 3 3.5 4 4.5 5 5.5 6 10
−910
−810
−710
−610
−510
−410
−310
−210
−1E
b/N
0
(dB)
BER
QC(2286,1914) LDPC code MBC−SLC,(255,239,2)BCH code MBC−SLC,(511,484,3)BCH code MBC−SLC,(1023,973,5)BCH code
Figure 4.8: Calculated BER performances of the MBC-SLC scheme for the QC(2286,1914) LDPC code concatenated with different BCH codes. Also shown is the performance of the LDPC code.
2 2.5 3 3.5 4 4.5 5 5.5 6 10
−810
−710
−610
−510
−410
−310
−210
−1E
b/N
0
(dB)
BER
QC(2286,1914) LDPC code MBC−SLC,(255,239,2)BCH code MBC−SLC,(511,484,3)BCH code MBC−SLC,(1023,973,5)BCH code
Figure 4.9: Simulated BER performances of the MBC-SLC scheme for the QC(2286,1914) LDPC code concatenated with different BCH codes. Also shown is the performance of the LDPC code.
3 3.5 4 4.5 5 10
−910
−810
−710
−610
−510
−410
−310
−2E
b/N
0
(dB)
BER
QC(4590,3835) LDPC code
MBC−SLC,(1023,1003,2)BCH code MBC−SLC,(2047,2003,4)BCH code
Figure 4.10: Calculated BER performances of the MBC-SLC scheme for the QC(4590,3835) LDPC code concatenated with different BCH codes. Also shown is the performance of the LDPC code.
3 3.5 4 4.5 5 10
−810
−710
−610
−510
−410
−310
−2E
b/N
0
(dB)
BER
QC(4590,3835) LDPC code
MBC−SLC,(1023,1003,2)BCH code MBC−SLC,(2047,2003,4)BCH code
Figure 4.11: Simulated BER performances of the MBC-SLC scheme for the QC(4590,3835) LDPC code concatenated with different BCH codes. Also shown is the performance of the LDPC code.
Chapter 5
Conclusion and Future Work
In this thesis, we analyze three concatenated coding schemes: The SBC-MLC scheme, the MBC-SLC scheme and the MBC-MLC scheme specifically in the error floor region under the premise that the BER decays very slow in this region. Given the error number distribution {Pi, ∀ 1 6 i 6 kLDPC} (obtained at a selective SNR in the error floor region) and assuming that it can be applied to other SNRs, our analysis and simulations show that the concatenated scheme can improve the error floor of the LDPC-only scheme. Yet, there is a restriction for these concatenated coding schemes. That is, a proper t must be chosen, where t is the error correcting capability of the BCH codes. From the performance standpoint, a larger t is often preferred, while a larger t may result in a larger rate loss (due to a different BCH code must be used) and hence compensate the gain from the concatenated scheme.
By investigating the MBC-SLC scheme for different BCH codes subject to a fixed system code rate, we found that a larger t is still a better choice. When comparing the SBC-MLC scheme with the MBC-SLC scheme, we would say that the SBC-MLC scheme has an obvi-ously larger improvement over its corresponding LDPC-only scheme, yet the improvement depends more on the selected t.
Unfortunately, such comparison between the SBC-MLC scheme and the MBC-SLC scheme
is rather weak in its footing since we could not exhaust all designs for the two schemes.
Hence, the following conclusion is drawn based more on intuition from simulations rather than on rigorous investigation: When the SBC-MLC scheme is composed of a BCH code with a stronger error correcting capability, its performance is often better than that of the MBC-SLC scheme (see for example Figures 5.1 and 5.2).
2 2.5 3 3.5 4 4.5 5 5.5 6
Figure 5.1: Comparison of calculated performances of the MBC-SLC scheme and the SBC-MLC scheme subject to the same system code rate. The red line corresponds to a con-catenated system consisting of eight (255,239,2) BCH codes and a QC(2286,1914) LDPC code. The black line corresponds to a concatenated system consisting of four (511,484,3) BCH codes and a QC(2286,1914) LDPC code. The pink line corresponds to a concatenated system consisting of two (1023,973,5) BCH codes and a QC(2286,1914) LDPC code. The red line with star marks corresponds to a concatenated system consisting of a (4095,3879,18) BCH code and two QC(2286,1914) LDPC codes.
3 3.5 4 4.5 5 10
−1410
−1210
−1010
−810
−610
−410
−2E
b/N
0
(dB)
BER
QC(4590,3835) LDPC code
MBC−SLC,(1023,1003,2)BCH code MBC−SLC,(2047,2003,4)BCH code SBC−MLC,(8191,7996,15)BCH code
Figure 5.2: Comparison of calculated performances of the MBC-SLC scheme and the SBC-MLC scheme subject to the same system code rate. The black line corresponds to a con-catenated system consisting of four (1023,1003,2) BCH codes and a QC(4590,3835) LDPC code. The pink line corresponds to a concatenated system consisting of two (2047,2003,4) BCH codes and a QC(4590,3835) LDPC code. The red line corresponds to a concatenated system consisting of a (8191,7996,15) BCH code and two QC(4590,3835) LDPC codes.
Our analysis is majorly dependent on the information of the error number distribution.
So our analysis can be applied to any system that can provide such information; hence, the same technique may be available for the BEC as well as BSC channels. Yet, if the error number distribution cannot be accurately estimated, or even cannot be applied to the SNR region other than the SNR at which such information is measured, our estimation formulas may fail. This is one of the problems that need to be resolved in the future. In addition, one may also wish to consider using the LDPC codes as the outer codes and the BCH codes as the inner codes. How such system performs and how to analyze it would be another future work of interest.
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