Decoding Algorithm and the Simulation Results
4.2 Simulation Results
In this section, we examine the codes proposed in Chapter 3 by simulations. In order to facilitate the comparison of our codes with the ones proposed in [1] and [2], we adopt the same system parameters therein. Specifically, channel coefficient h obeys zero-mean complex-Gaussian law with E[hhH] = (1/P )IP and P = 2, where IP denotes the P -by-P identity matrix. The average system SNR is given by
N
There will be three kinds of decoders used in our simulations. For simplify, we will adopt the following abbreviations, ‘ML’, ‘SP’ and ‘SNP,’ where they respectively stand for the maximum-likelihood (ML) decoder, the sequential (S) decoder introduced in Section
4.1 with heuristic prediction (P) function defined in (2.9), and the sequential (S) decoder introduced in Section 4.1 but with no heuristic prediction (NP) function.
As often done in literature [2], the decoding complexity is measured by the average path expansions per information bit. This complexity measure is conventionally considered to be proportional to the average number of metric calculations per information bit. In our case, however, such a convention is not exact because some expansions may require only one metric calculation while others need two metric calculations. Since the memory access such as insertion and deletion of path elements is in a sense a more significant key factor for decoding efficiency in nowadays computer technology, the adopted complexity measure is justified from the stack maintenance aspect.
Figures 4.1 and 4.2 compare our codes with the codes in [1] and [2]. The performance and complexity are both provided.
Then we examine the proposed codes of different codeword lengths. Specifically, Fig-ures 4.3-4.7 are for codes of codeword length 18. FigFig-ures 4.8-4.12 are for codes of codeword length 20. Figures 4.13-4.17 are for codes of codeword length 22. Figures 4.18-4.22 are for codes of codeword length 24. And Figures 4.23-4.27 are for codes of codeword length 26.
For each length, we examine the proposed codes for different maximum state numbers and using different decoders. This results in five plots for each codeword length. The first plot shows the difference of word error rates (WER) among codes with different maximum state numbers. The second plot shows the complexity of the SP decoder for the codes examined in the first plot. The third plot turns to the complexity of the SNP decoder. And the fourth and fifth plots respectively compare the performance and complexity among different decoders.
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 10−4
10−3 10−2 10−1 100
SNR (dB)
WER
SA(ML)
SOTS(SP), SOTS−2048(SP) SOTS−512
SOTS−128
Figure 4.1: Word error rates (WERs) for the computer-searched code found by simulated an-nealing in [1] (SA(ML)), the SOTS code with single code tree (SOTS(SP)) in [2] which is also the code with maximum state number 2048 (SOTS-2048(SP)), the code with maximum state number 512 (SOTS-512(SP)), and the code with maximum state number (SOTS-128(SP)).
The SA(ML) code is decoded by the ML decoder, while all the other codes are decoded by the SP decoder. The code rate is 1/2 and the codeword length is N = 22.
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 100
101 102 103 104
SNR (dB)
Average numbers of path expansions per information bit
SA(ML)
SOTS(SP), SOTS−2048(SP) SOTS−512
SOTS−128
Figure 4.2: The average numbers of path expansions per information bit for the codes examined in Figure 4.1.
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 10−3
10−2 10−1 100
SNR (dB)
WER
SOTS−128(SP) SOTS−256(SP) SOTS−512(SP)
Figure 4.3: Word error rates (WERs) for the self-orthogonal trellis-structure codes of length N = 18 with maximal state number 128, 256 and 512, respectively. The decoders used are the sequential decoding with heuristic prediction (SP). The code rate is 1/2.
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 100
101 102
SNR (dB)
Average number of path expansions per information bit
SOTS−128(SP) SOTS−256(SP) SOTS−512(SP)
Figure 4.4: The average numbers of path expansions per information bit for the codes examined in Figure 4.3.
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 101
102 103
SNR (dB)
Average number of path expansions per information bit
SOTS−128(SNP) SOTS−256(SNP) SOTS−512(SNP)
Figure 4.5: The average numbers of path expansions per information bit for the codes exam-ined in Figure 4.3 except the codes are now decoded by the sequential decoding algorithm with no heuristic prediction (SNP).
0 5 10 15
Figure 4.6: Word error rates (WERs) for the proposed codes respectively with S = 128, 256 and 512, decoded by different approaches: ML, SP and SNP. The code rate is 1/2 and the codeword length is N = 18.
0 5 10 15
Figure 4.7: The average numbers of path expansions per information bit for the codes examined in Figure 4.6.
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 10−3
10−2 10−1 100
SNR (dB)
WER
SOTS−128(SP) SOTS−256(SP) SOTS−512(SP) SOTS−1024(SP)
Figure 4.8: Word error rates (WERs) for the self-orthogonal trellis-structure codes of length N = 20 with maximal state number 128, 256, 512 and 1024, respectively. The decoders used are the sequential decoding with heuristic prediction (SP). The code rate is 1/2.
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 100
101 102
SNR (dB)
Average number of path expansions per information bit
SOTS−128(SP) SOTS−256(SP) SOTS−512(SP) SOTS−1024(SP)
Figure 4.9: The average numbers of path expansions per information bit for the codes examined in Figure 4.8.
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 101
102 103
SNR (dB)
Average number of path expansions per information bit
SOTS−128(SNP) SOTS−256(SNP) SOTS−512(SNP) SOTS−1024(SNP)
Figure 4.10: The average numbers of path expansions per information bit for the codes ex-amined in Figure 4.8 except the codes are now decoded by the sequential decoding algorithm with no heuristic prediction (SNP).
0 5 10 15
Figure 4.11: Word error rates (WERs) for the proposed codes respectively with S = 128, 256, 512 and 1024, decoded by different approaches: ML, SP and SNP. The code rate is 1/2 and the codeword length is N = 20.
0 5 10 15
Figure 4.12: The average numbers of path expansions per information bit for the codes examined in Figure 4.11.
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 10−3
10−2 10−1 100
SNR (dB)
WER
SOTS−128(SP) SOTS−256(SP) SOTS−512(SP) SOTS−1024(SP) SOTS−2048(SP)
Figure 4.13: Word error rates (WERs) for the self-orthogonal trellis-structure codes of length N = 22 with maximal state number 128, 256, 512, 1024 and 2048, respectively. The decoders used are the sequential decoding with heuristic prediction (SP). The code rate is 1/2.
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 100
101 102 103
SNR (dB)
Average number of path expansions per information bit SOTS−128(SP)
SOTS−256(SP) SOTS−512(SP) SOTS−1024(SP) SOTS−2048(SP)
Figure 4.14: The average numbers of path expansions per information bit for the codes examined in Figure 4.13.
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 101
102 103
SNR (dB)
Average number of path expansions per information bit SOTS−128(SNP)
SOTS−256(SNP) SOTS−512(SNP) SOTS−1024(SNP) SOTS−2048(SNP)
Figure 4.15: The average numbers of path expansions per information bit for the codes exam-ined in Figure 4.13 except the codes are now decoded by the sequential decoding algorithm with no heuristic prediction (SNP).
0 5 10 15
Figure 4.16: Word error rates (WERs) for the proposed codes respectively with S = 128, 256, 512, 1024 and 2048, decoded by different approaches: ML, SP and SNP. The code rate is 1/2 and the codeword length is N = 22.
0 5 10 15
Figure 4.17: The average numbers of path expansions per information bit for the codes examined in Figure 4.16.
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 10−4
10−3 10−2 10−1 100
SNR (dB)
WER
SOTS−128(SP) SOTS−256(SP) SOTS−512(SP) SOTS−1024(SP) SOTS−2048(SP) SOTS−4096(SP)
Figure 4.18: Word error rates (WERs) for the self-orthogonal trellis-structure codes of length N = 24 with maximal state number 128, 256, 512, 1024, 2048 and 4096, respectively. The decoders used are the sequential decoding with heuristic prediction (SP). The code rate is 1/2.
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 100
101 102 103
SNR (dB)
Average number of path expansions per information bit
SOTS−128(SP) SOTS−256(SP) SOTS−512(SP) SOTS−1024(SP) SOTS−2048(SP) SOTS−4096(SP)
Figure 4.19: The average numbers of path expansions per information bit for the codes examined in Figure 4.18.
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 101
102 103 104
SNR (dB)
Average number of path expansions per information bit
SOTS−128(SNP) SOTS−256(SNP) SOTS−512(SNP) SOTS−1024(SNP) SOTS−2048(SNP) SOTS−4096(SNP)
Figure 4.20: The average numbers of path expansions per information bit for the codes exam-ined in Figure 4.18 except the codes are now decoded by the sequential decoding algorithm with no heuristic prediction (SNP).
0 5 10 15
Figure 4.21: Word error rates (WERs) for the proposed codes respectively with S = 128, 256, 512, 1024, 2048 and 4096, decoded by different approaches: ML, SP and SNP. The code rate is 1/2 and the codeword length is N = 24.
0 5 10 15
Figure 4.22: The average numbers of path expansions per information bit for the codes examined in Figure 4.21.
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 10−3
10−2 10−1 100
SNR (dB)
WER
SOTS−256(SP) SOTS−512(SP) SOTS−1024(SP) SOTS−8192(SP)
Figure 4.23: Word error rates (WERs) for the self-orthogonal trellis-structure codes of length N = 26 with maximal state number 256, 1024 and 8192, respectively. The decoders used are the sequential decoding with heuristic prediction (SP). The code rate is 1/2.
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 100
101 102 103
SNR (dB)
Average number of path expansions per information bit
SOTS−256(SP) SOTS−512(SP) SOTS−1024(SP) SOTS−8192(SP)
Figure 4.24: The average numbers of path expansions per information bit for the coded examined in Figure 4.23.
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 102
103 104
SNR (dB)
Average number of path expansions per information bit
SOTS−256(SNP) SOTS−512(SNP) SOTS−1024(SNP) SOTS−8192(SNP)
Figure 4.25: The average numbers of path expansions per information bit for the codes exam-ined in Figure 4.23 except the codes are now decoded by the sequential decoding algorithm with no heuristic prediction (SNP).
0 5 10 15
Figure 4.26: Word error rates (WERs) for the proposed codes respectively with S = 256, 512, 1024 and 8192, decoded by different approaches: ML, SP and SNP. The code rate is 1/2 and the codeword length is N = 26.
0 5 10 15
Figure 4.27: The average numbers of path expansions per information bit for the codes examined in Figure 4.26.