Performance in AWGN Channel with Floating-Point Processing

5.1.1 Number of Iterations

The iteration number is a most important factor in the decoding algorithm. This number affects the decoding accuracy and system complexity. A larger iteration number usually leads to better performance. But the complexity and the and the latency are increased. We compare the performance with iteration numbers between 10 and 70 for BP decoding of the rate 1/2, length 576 code with QPSK modulation in Fig. 5.1.

In Fig. 5.1, the performance at 10 iterations is obviously inferior to other choice. We can see that if the iteration number is more than 20, then the BER curves are almost the same.

To limit the decoding complexity and maintain a reasonable performance, we use 20 as the iteration number in other simulations.

1 1.5 2 2.5 3 3.5

LDPC code BP decoding with Different Iteration, Rate 1/2, Length 576, QPSK

Iteration 10

Figure 5.1: LDPC decoding performance in different iteration numbers with floating-point computation.

5.1.2 Performance at Different Codeword Lengths

In convolutional coding the codeword length does not affect the performance. But LDPC code is different. Fig. 5.2 shows the performance at four different codeword lengths at code rate 1/2 with QPSK modulation and BP decoding with 20 iterations. In Fig. 5.2, as the codeword length becomes longer, the improved performance is obtained. The result is not surprising, because at medium or short code lengths, the BP algorithm is not optimum, owing to correlation among messages passed during iterative decoding [17]. For the codeword length 2304, the coding gain is about 8 dB at BER 10⁻⁶. This coding gain value is several dB higher than convolutional coding as obtained in the last chapter.

5.1.3 Performance with Different Modulations

We compare the performance of QPSK, 16QAM and 64QAM at rate 1/2, codeword length 576, with BP decoding with 20 iterations. In Fig. 5.3, the coding gains of QPSK, 16QAM,

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LDPC code BP decoding with Different Length, Rate 1/2, Iteration 20, QPSK

Uncoded QPSK Length 576 Length 1152 Length 1728 Length 2304

Figure 5.2: LDPC decoding performance in different codeword length with floating-point computation.

LDPC code BP decoding with Different Modulation, Rate 1/2, Length 576, Iteration 20

Uncoded QPSK

Figure 5.3: LDPC decoding performance with different modulation employing floating-point computation.

Table 5.1: Comparison of Coding Gain Between LDPC Codes and Convolutional Codes at Code Rate 1/2 in AWGN at BER = 10⁻⁶

Modulation Type Convolutional Code LDPC Code

QPSK 5.62 7.31

16QAM 6.28 7.43

64QAM 6.35 9.32

64QAM are 7.31, 7.43 and 9.32 dB, respectively, at BER=10⁻⁶.

In Table 5.1, we compare the coding gains of LDPC codes and convolutional codes. The LDPC codes are obviously better. They are close to Shannon limit [12].

5.1.4 Performance at Different Coding Rates

In IEEE 802.16e, six coding rate are defined for LDPC code, namely ¹₂, ²₃A, ²₃B, ³₄A, ³₄B and ⁵₆. In Fig. 5.4, we compare their performance under QPSK, code length 576 and BP decoding with 20 iteration. As mat be anticipated, the best performance is obtained at rate 1/2. As the code rate gets higher, the performance gets worse. Note that the two curves of ²₃A and ²₃B are very close, but still have some difference. We can explain why this little difference exists from the point of view of “threshold” [32]. As the block length tends to infinity, an arbitrarily small bit error probability can be achieved if the noise level is smaller than a certain threshold. In Table 5.2, the threshold of ²₃A is only larger than that of ²₃B by 0.012 dB. So, the BER curves are very close, and the curve for ²₃A is a little better than that of ²₃B. In our simulation, these two curves really follow the threshold analysis. By the similar method, we also easily explain the relationship between the two BER curves of ³₄A and ³₄B.

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LDPC code BP decoding with Different Coding Rate, QPSK, Length 576, Iteration 20

Rate 1/2

Figure 5.4: LDPC Decoding Performance in Different Coding Rate (floating-point).

Table 5.2: Threshold for Each Code Rate under BPSK Modulation in AWGN Channel [20].

Code Rate Threshold

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LDPC code decoding with Different Algorithm, QPSK, Rate 1/2, Length 576, Iteration 20

BP Decoding Min−Sun Decoding

Normalized BP−Based Decoding with α=1.125 Offset BP−Based Decoding with β=0.125

Figure 5.5: LDPC decoding performance using different decoding algorithm employing floating-point computation.

5.1.5 Performance of Reduced-Complexity Algorithm

In chapter 2, we discuss some decoding algorithms with reduced complexity than the BP algorithm. In Fig. 5.5, we compare the performance of four algorithms at code rate 1/2, length 576, QPSK modulation with 20 iterations.

As expected, the min-sun algorithm is obviously worse than the other algorithms. The reason also been discussed previously in chapter 2.

The other two reduced-complexity algorithms, offset BP-based and normalized BP-based, have even a slightly better performance than the BP algorithm. These results are not sur-prising, because at medium or short code lengths, the BP algorithm is not optimum. This is because the number of short cycles in their Tanner graphs influences the BP decoding performance which depends on the amount of correlation between messages, and the two reduced-complexity BP-based algorithms seem to outperform the BP algorithm by reduc-ing the negative effect of the correlations [17]. The normalized BP-based algorithm slightly outperforms the offset BP-based algorithm, but may also be slightly more complex to

im-Table 5.3: LDPC Coding Gain between Floating-point and Fixed-point in AWGN at BER

As a result, we choose the offset BP-Based algorithm for DSP implementation. The structure of the algorithm also makes the conversion from floating-point to fixed-point com-putation easier.

5.2 Performance in AWGN Channel with Fixed-Point