In this thesis, we first utilize a (2, 1, 6) convolutional codes, g = [163 135]octal and a generator polynomial g(x) = 1+x2+x4+x6+x7+x8 for CRC-8. Here, we remain the original CRC encoding scheme in this simulation. If we ignore four bits of the register of CRC-8, namely, we use the combined trellis of CRC-4 rather than the combined trellis of CRC-8 for decoding. Since the combined trellis of the specifical CRC-4 is one of the sub-trellis of the reduced trellis of the combined trellis of CRC-8, it is reasonable to employ the combined trellis of CRC-4 directly for decoding. Noted that the generator polynomial of the CRC-4 is g(x) = 1 +x2+x4which is the left half part of the genertor polynomial of the CRC-8 we use.
It is explicit that the proposed algorithm is 0.7dB superior to e conventional convolutional decoding which only employs CC for decoding in Figure 4.1. The improvement on bit error rate is 0.6dB greater than the conventional one as illustrated in Figure 4.2. The combined CRC-8 ML is the line which is the ML performance of the concatenated code comprised of CRC and CC. Now consider about the complexity of the proposed algorithm. In Table 4.1, the proposed algorithm has ten times the number of computation times of the conventional decoding algorithm in low SNR (1-1.5dB). Nevertheless, the number of computation times of these two schemes are similar in medium SNR (3-4dB) and the proposed algorithm is even greater than the conventional one in high SNR (>4dB) due to the CRC merging stage in the proposed algorithm. More specifically, the CRC merging stage in the end of the reduced version of the combined trellis can extract some unneeded steps because of the elimination
-1 0 1 2 3 4 5 10-5
10-4 10-3 10-2 10-1 100
CC (2,1,6), Infor Size=114, CRC-8 detect
Block Error Rate
SNR CConly CRC-8
CRC-8 ignore 4bit last Merge combined CRC-8 ML
Figure 4.1: CC (2,1,6), information size=114, CRC-8 detect
of paths caused by CRC constrain. Therefore, the number of computation times can be decreased in the latter portion of the combined trellis when the noise is slight.
The comparison between the former idea of partition the CRC bits and the proposed decoding algorithm is shown in Figure 4.6. We utilize a (2, 1, 6) convolutional codes, g = [163 135]octaland a generator polynomial g(x) = 1+x3+x4+x7+x8+x10+x14+x15for CRC-15. In Figure 4.3, there are three CRC encoding schemes that we apply in this comparison:
the first is the conventional CRC encoding for CRC-15. In the second, we partition CRC-15 into the error correction section which is CRC-8 and the error detection section which is CRC-7. Partition CRC-15 into two CRC-4 for error correction and one CRC-7 for error detection in the third encoding scheme. The particular description for the second and third methods is exhibited in Chapter 4. Note that the generator polynomial of 7 and CRC-8 is g(x) = 1 + x3 + x7 and g(x) = 1 + x2+ x4+ x6 + x7+ x8 respectively. As illustrated in Figure 4.6, the performance of the proposed algorithm of ignoring 7 bits for CRC-15 is similar to the second scheme in 4.3. However, the proposed algorithm preserves the original CRC encoding scheme but still improves the performance. In addition, on account of using
-1 0 1 2 3 4 5 10-6
10-5 10-4 10-3 10-2 10-1 100
CC (2,1,6), Infor Size=114, CRC-8 detect
Bit Error Rate
SNR
CConly CRC-8
CRC-8 ignore 4bit last Merge combined CRC-8 ML
Figure 4.2: CC (2,1,6), information size=114, CRC-8 detect
CRC-8+CC ignore 4bits CC only
SNR Average Maximum Average Maximum
1 151222 223044 12702 14974
1.5 116265 222900 11789 14974
2 75132 222614 10691 14974
2.5 40269 223068 9020 14974
3 17216 210186 7108 14974
3.5 6641 182972 5007 14974
4 2262 202466 3068 14974
4.5 933 189084 1652 14910
5 478 112218 844 13762
Table 4.1: complexity
114 8 7
114 15
57 4 57 4 7
Figure 4.3: Three distinct encoding schemes for CRC codeword
the identical register of CRC-4 to form the combined trellis for the proposed algorithm and the partitioned combined trellis for the second scheme in 4.3, the complexity of the second scheme is slightly fewer than the the complexity of the proposed algorithm because there are two merging stage in the trellis for second scheme. Also, the second scheme approach a superior error rate to the proposed decoding algorithm which uses unaltered encoding scheme. We conclude that the performance can be upgraded by change the CRC encoding scheme.
Now we compare the LVA to our proposed decoding algorithm in Figure 4.5. The block error rate of the proposed decoding algorithm is equivalent to the error rate of the LVA with L=10 in SNR 0-2.5dB and is between the error rate of LVA with L=2 and L=10 in SNR
>3dB.
In the retransmission system which permitting once retransmission and employing Chase combining, the proposed decoding algorithm is still superior to the conventional decoding algorithm about 0.7-0.8dB. Note that the codeword decoded from the combined trellis of CRC-8 and CC will always pass the CRC check, so the combined CRC-8 ML has no simu-lations for retransmission.
The throughput of the proposed decoding algorithm is similar to the throughput of the LVA with L=10 and is greater than the conventional decoding algorithm.
0 1 2 3 4 5 10-5
10-4 10-3 10-2 10-1 100
CC (2,1,6), Infor Size=114, CRC-7 detect
Block Error Rate
SNR CConly L1 CRC-15 CRC-CC p1L1 CRC-g8 G7 CRC-CC p2L1 CRC-g4g4 G7 CRCignore11 4lastMerge CRC-15 CRCignore7 8lastMerge CRC-15
Figure 4.4: CC (2,1,6), information size=114, CRC-15 detect
-1 0 1 2 3 4 5
10-5 10-4 10-3 10-2 10-1 100
CC (2,1,6), Infor Size=114, CRC-8 detect
Block Error Rate
SNR
CConly CRC-8 CConly CRC-8 L =2 CConly CRC-8 L =10 CRC-8 ignore 4bit last Merge combined CRC-8 ML CConly CRC-8 retrans. Once CRC-8 ignore4bit retrans. Once
Figure 4.5: CC (2,1,6), information size=114, CRC-8 detect
-1 0 1 2 3 4 5
CC (2,1,6), Infor Size=114, CRC-8 detect
Throughput
SNR
CConly CRC-8 CConly CRC-8 L =2 CConly CRC-8 L =10 CRC-8 ignore 4bit last Merge combined CRC-8 ML CConly CRC-8 retrans. Once CRC-8 ignore4bit retrans. Once
Figure 4.6: Throughput: CC (2,1,6), information size=114, CRC-15 detect
-6 -4 -2 0 2 4 6
CC (2,1,6), Infor Size=114, CRC-8 detect
Block Error Rate
SNR
CConly CRC-8 CConly CRC-8 L =2 CConly CRC-8 L =10 CRC-8 ignore 4bit last Merge combined CRC-8 ML CConly CRC-8 retrans. Once CRC-8 ignore4bit retrans. Once CConly CRC-8 retrans. TWICE CRC-8 ignore4bit retrans. TWICE CConly CRC-8 retrans. 3times CRC-8 ignore4bit retrans. 3times retrans. 3 times
retrans. twice
retrans. once
Figure 4.7: CC (2,1,6), information size=114, CRC-8 detect
Chapter 5 Conclusions
The conventional decoding scheme in hybrid ARQ utilizes only convolutional codes to de-code the received sequence and considers CRC as just an error detecting de-code. However, the proposed decoding algorithm combine the error correcting ability of CRC with the con-volutional decoding to alleviate the error rate caused by concon-volutional decoder. Due to the effect of the CRC constrain which added in the combined trellis, the codeword decoded from the proposed decoding algorithm has more opportunities to pass the error detection.
There is a decoding algorithm, list Viterbi algorithm, which can also decrease the error rate generated from the CC decoder by attaining Lth best paths in 1st transmission instead of one ML path. The LVA tries to search another acceptable decoded codeword to pass the error detection before next retransmission while the ML codeword failed. The combination of the LVA and our proposed algorithm can achieve substantial improvement in the perfor-mance. Instead of employ the combined trellis of a CRC with a large degree, we can use the reduced version of the combined trellis and the LVA to upgrade the performance without tremendous computation times.