Low-density Parity-check Code

LDPC codes are linear block codes originally invented by Robert Gray Gallager [Gal62], which are very powerful codes and can operate close to the Shannon capacity [RS+_01]. The special property of them is the sparse parity check matrix, which describes the code completely. A distinction is generally made between regular and irregular LDPC codes. A regular LDPC code fulfills the condition that the number of ones in each column and the number of ones in each row of the parity check matrix are identical. Typically, irregular LDPC codes have a higher performance than regular LDPC codes [RS+_{01]. As exemplary}

P =      0 1 0 1 1 0 0 1 1 1 1 0 0 1 0 0 0 0 1 0 0 1 1 1 1 0 0 1 1 0 1 0     

x

x

x

x

x

x

x

x

p

p

p

p

Figure 4.6: Parity check matrix and Tanner graph representation of a regular (8, 4)

LDPC code.

shown in Fig. 4.6, the parity check matrix can be visualized by a bipartite Tanner graph consisting of variable nodes xi and check nodes pi. A “1” in the parity check matrix

symbolizes a connection between a variable node and a check node. The check nodes can be interpreted as individual SPC codes which represent the parity checks. The strong potential of the LDPC codes lies in their iterative soft-input soft-output decoding. According to the turbo principle, extrinsic information can be exchanged between variable nodes and check nodes using message-passing algorithms in order to increase decoding performance, which will be covered in Chapter 5. In this dissertation, the suboptimal graph-based belief- propagation algorithm [MM+_{98] is used for soft-input decoding. In order to guarantee a}

long exchange of independent messages, the length of circles in the parity check matrix should be kept as large as possible. A girth is defined as the circle with the minimum length. It limits the code performance considerably [Joh10]. In the example in Fig. 4.6, there is a girth of length four, which is highlighted in red in the parity check matrix and the Tanner graph.

In diffusion-based molecular communication, LDPC codes are used for the first time in [LH+_{15a] together with a one step majority logic decoding. As in [DS}+_{19], the focus in this} dissertation is on the advanced LDPC codes introduced in the second standard of digital video broadcasting over satellite (DVB-S2) [DVBS2]. As shown in Tab. 4.4, LDPC codes

Table 4.4: LDPC parameters from the DVB-S2 standard [DVBS2].

nc 64800

R 1/4 1/3 2/5 1/2 3/5 2/3 3/4 4/5 5/6 8/9 9/10

nc 16200

R 1/5 1/3 2/5 4/9 3/5 2/3 11/15 7/9 37/45 8/9

are given with rates from R =1_{/4 to R =}9_{/10 for moderate block length and from R =}1_/5 to R = 8_{/9 for short block length. A short block length is defined as nc = 16200 and a} moderate block length as nc= 64800. Here, only the LDPC codes with the moderate block length nc = 64800 are considered. For encoding, all parity bits are initialized with zero and determined according to the construction rules and tables given in the DVB-S2 standard. After their determination, the parity bits are accumulated. Since they are systematic LDPC codes, the info bits are concatenated with the parity bits, which results in the final code word.

Numerical Results

Fig. 4.7 shows the influence of the transmission distance on the BER performance of LDPC coded transmissions under different normalizations. For the analysis, a low-rate R =1_/4 LDPC code, a medium-rate R =1_{/2 LDPC code, and a high-rate R =}9_{/10LDPC code are} considered. Compared to the simpler CRC, SPC, repetition, Hamming, and Reed-Solomon codes from the previous sections, the LDPC codes achieve significantly higher coding gains for comparable rates. In addition, they show a steep slope to the lower BER regions, known as turbo cliff. The reason is the larger code word length of the LDPC codes and their more complex structure, which enables iterative decoding with information exchange between variable nodes and parity check nodes. In the results shown in Fig. 4.7, the maximum

4.1 Block Codes 10 15 20 25 30 35 40 45 50 10−6 10−5 10−4 10−3 10−2 10−1 d in µm BER APP R =9/10 R=1/2 R=1/4 (a) Unnormalized. 10 15 20 25 30 35 40 45 50 10−6 10−5 10−4 10−3 10−2 10−1 d in µm BER (b) T normalized. 10 15 20 25 30 35 40 45 50 10−6 10−5 10−4 10−3 10−2 10−1 d in µm BER (c) T & N normalized.

Figure 4.7:Bit error rate performance of LDPC encoded transmissions as a function

number of internal iterations is limited to 50 iterations.

Fig. 4.7(a) shows the unnormalized case. LDPC codes with lower code rates achieve a larger coding gain due to their increased redundancy. With a target BER of 10−3_{, the} R =1/2 LDPC code already achieves an increase in the maximum transmission distance from 28 µm to 44 µm. The R =1_{/4LDPC code further increases the maximum transmission} rate to 69 µm which can be observed in Fig. 5.7(a).

In Fig. 4.7(b), the symbol duration is normalized to the information rate. As described previously, this has a negative effect on the BER performance due to shorter symbol dura- tions. The lower the code rate, the larger the negative effect. Thus, the BER performance of the R =9_{/10 LDPC code changes only slightly compared to the unnormalized analysis.} In contrast to CRC, SPC, repetition, Hamming, and Reed-Solomon codes discussed in the previous sections, the general performance behavior of the LDPC codes with respect to the unnormalized case does not reverse. In Fig. 4.7(b), low-rate LDPC codes are still more powerful than high-rate LDPC codes. This shows that in the scenario under consideration the coding gain of LDPC codes by additional redundancy exceeds the limitations of shorter symbol durations. With a target BER of 10−3_{, the R =1/4LDPC code achieves a maximum} transmission distance of 51 µm, which can be observed in Fig. 5.7(a).

Finally, Fig. 4.7(c) shows in addition to the symbol duration the normalization of the average number of released molecules per info bit. As described in previous sections, this leads to a reduction in signal strength, which has a stronger effect on lower-rate codes. The combination of these two negative effects (reduction of symbol duration and number of molecules) dominates the positive effect (coding gain through additional redundancy). Consequently, higher-rated LDPC codes achieve better BER performances than lower-rated LDPC codes. Despite the normalizations, the R =9_{/10LDPC code increases the maximum} transmission distance at a target BER of 10−3 _{to 31 µm.}