As the advisable overheads of sub-codes are obtained, the UEP properties of the proposed coding scheme should be determined by the output degree distributions of sub-codes.
Consider each source node Si, to obtain priority disparity Ki, the output degree distributionΩ(i)(x) should satisfies
Ω(i)(1)
γi=KiγΦ(1), (9) whereγis the overall overhead, and the left part of Eq. (9) is the average degree of the input symbols on source node Si after the first encoding step.
After the second step of the encoding process, the average degree of input symbols on each source node Siare increased by times (Ω(R)(1)), but for each in put symbol, the number of its identity neighbors has not been increased. In other word, although the average degrees of input symbols increased after the second step of encoding process, but all the output neighbors of each input symbol are also been the neighbors of the intermediate symbols which are connected with this input symbol. For this reason, the LT encoder on relay node was not implemented to improve the error performance but to overcome the erasure probability of the relay channel. For this reason, and consider the overall compute complexity, the LT encoder on relay node should be assigned output degree distributionΩ(R)(x) with low average degree.
As the Robust degree distributions of LT codes can provide nearly optimal decoding performances, and the Robust degree distribution is determined by the variablesk, δandc, where δis the allowable decoding failure probability andc is a constant, then by assign different value toδandc, one can obtain different Robust degree distribution. Hence, for the source nodes, the degree distributions should satisfy Eq. (9), and for the relay node.
5 Simulation Results
In this section, we first take the asymptotic and finite length evaluation of pro- posed codes, then the comparisons between proposed codes and conventional distributed UEP codes are also given.
Consider a proposed code with two LEOs and single GEO, where the number of input symbols and overhead of sub-codes on LEOs are the same, and output degree distributions for sub-codes on LEOs areΩ(1)(x) = 0.007969x1+ 0.493570x2 + 0.166220x3 + 0.072646x4 + 0.082558x5 + 0.056058x8 + 0.037229x9 + 0.055590x19+ 0.025023x64 + 0.003137x66andΩ(2)(x) = 0.0782x+ 0.4577x2+ 0.1706x3 + 0.0750x4 + 0.0853x5 + 0.0376x8 + 0.0380x9 + 0.0576x19,respec- tively. The output degree distribution for sub-code on GEO isΩ(R)(x) = 0.057x+ 0.4589x2 + 0.17x3 + 0.1156x4 + 0.0754x5 + 0.0575x6 + 0.0382x7 + 0.0274x8, and the overhead on GEO is 1.05. The asymptotic error performance of the pro- posed code is shown in Fig.4, where the input symbols on LEO S1can provide better error performance than which on S2, which means the proposed code can provide UEP property between input symbols on different LEOs. Figure5shows the finite length error performances of proposed codes withk1=k2= 10000 and k1=k2= 1000, it is easy to say the overhead and error performances of proposed codes would as better as larger number of input symbols on source nodes.
0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5
10−4 10−3 10−2 10−1 100
overhead
symbol error rate
LEO S1 LEO S2
Fig. 4.Asymptotic error performance of proposed code with 2 source nodes and single relay node.
Then we make the comparison between proposed codes and conventional distributed UEP codes. Assume the sub-codes on LEOs of conventional code with the same output degree distribution, which is same asΩ(1)(x), and the sub-code on GEO also share the same output degree distribution as the proposed code.
Different with the proposed codes, the UEP property of conventional codes are mainly determined by the sub-code on relay nodes, then we assume the overhead of both sub-codes on source nodes are 1.05. To compare fairly, the conventional code has been assigned priority disparity KM = 1.7, then the proposed code
0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 10−3
10−2 10−1 100
overhead
symbol error rate
S1, Asymptotic S2, Asymptotic S1, k1=10000 S2, k2=10000
Fig. 5.Finite length error performance of proposed code with two source nodes and single relay node.
0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5
10−4 10−3 10−2 10−1 100
overhead
symbol error rate
Proposed code S1, k1=10000 Proposed code S2, k2=10000 Conventional code S1, k1=10000 Conventional code S2, k2=10000
Fig. 6.Finite length error performance of proposed code and conventional distributed UEP rateless code.
and conventional codes would provide the same UEP properties at the finite length condition where the input symbols on both source nodes are k1 =k2= 10000,its can be found in Fig.6, the proposed code can provide better overhead performance than conventional codes, which because of the drawback of the LT-based UEP codes.
6 Conclusion
In this paper, we propose a new class of distributed UEP rateless codes which can be using transmit multi kinds of data with different reliable requirements on satellite networks. All the sub-codes of proposed UEP rateless codes are EEP LT codes, hence the proposed code can provide better overhead property than the conventional distributed UEP rateless codes. As the relay node (GEO) in proposed code provide EEP property, which means the relay node not have to know the reliable requirements of input symbols on different source nodes, hence the security of input symbols can be ensured. We also derive the asymptotic and finite-length analysis of proposed codes. And the numerical results shown the proposed can provide the same UEP property as conventional distributed UEP rateless codes with low overhead performance.
Acknowledgment. This work was supported by National Natural Science Founda- tion of China. (No. 61601147, No. 61571316, No. 61371100) and “the Fundamental Research Funds for the Central Universities” (Grant No. HIT. MKSTISP. 2016013).
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Rateless Codes with Equal Recovery Time Property
Shuang Wu1, Zhenyong Wang1,2(B), Dezhi Li1, Gongliang Liu1, and Qing Guo1
1 School of Electronics and Information Engineering, Harbin Institute of Technology, Harbin 150001, China wooshuang@126.com, {zywang,lidezhi,liugl,qguo}@hit.edu.cn
2 Shenzhen Academy of Aerospace Technology, Shenzhen 518057, China
Abstract. A new class of rateless codes which are able to provide unequal error protection (UEP) and equal recovery time (ERT) prop- erties is proposed in this paper. Existing UEP-based LT codes have an important property termed unequal recovery time (URT), which means the data with different reliability requirements can be recovered with different overhead, and it is worth noting that the most important bits (MIB) also have better recovery time performance. The proposed codes can recover data with the same overhead and different error performance.
We analyze the asymptotic and experimental error performance of the proposed codes, and give the comparison between the proposed and tra- ditional codes, our results show that the new class of UEP rateless codes are useful for scenarios in which the data have different reliability and same timeliness requirements.
Keywords: Unequal error protectionãAsymptotic analysis Finite-length analysisãEqual recovery timeãRateless codes
1 Introduction
LT codes, the first practical codes of the family named rateless codes, were invented by Luby [1]. The code rate of rateless codes are not fixed, in other words, the output symbols can be generated as many as needed.
Rateless codes could also provide an important property which named unequal error protection (UEP). The UEP rateless codes proposed by Rah- navard in [2,3] by distribute different selection probabilities to input symbols in different blocks in the encoding process, Sejdinovic etc. construct the UEP rateless codes by dividing the data into a series windows, and different window have different encoding times [4]. The mentioned schemes are all based on single source, Talari and Rahnavard also proposed a coding scheme by using distributed rateless codes to fit for two source one relay scenario and provide UEP prop- erty [5,6]. The UEP rateless codes also be used to solve some practical scenarios where the different data have different reliability requirements.
c ICST Institute for Computer Sciences, Social Informatics and Telecommunications Engineering 2018 X. Gu et al. (Eds.): MLICOM 2017, Part I, LNICST 226, pp. 574–583, 2018.
https://doi.org/10.1007/978-3-319-73564-1_57
All the above mentioned UEP rateless codes have the same property which named unequal recovery time (URT). As for UEP rateless codes, the data which have better error performance, they always can be recovered faster, and the oth- ers are slower. In the ground transmission systems, as each entire encoding and decoding process only need a very short time slot duration, the URT property is negligible. Therefore, the URT property always be considered as by product, but for some scenarios where the duration time of each entire encoding and decoding process must be concerned, (for example, the deep space data trans- mission systems), the URT property may influence the user experience. Aiming to solve this problem, we proposed a new class of rateless codes which can pro- vide UEP property and the recovery time of each block is nearly same. The proposed codes could provide UEP property, but as the data in different parts have different error protection level, these data could be recovered nearly at the same time, in other words, this cods could provide equal recovery time (ERT) property. For the proposed LT codes in this paper, the MIB and LIB parts would be recovered nearly at the same time, then the overall timeliness property would be better than the mentioned UEP/URT LT codes.
The paper is organized as follows. In Sect.2, We review the related works, including the And-Or Tree analysis and the UEP cods [3]. The codes we proposed which could provide UEP and ERT properties are introduced in Sect.3, a simple example and its asymptotic performance analysis are also given. Section4shows the comparison between the proposed UEP/ERT LT codes and the comparative UEP/URT LT codes by asymptotic and experimental results. And the conclusion of this paper is drawn in Sect.5.
2 Related Works
In this section, we review the coding scheme proposed in [3] and analyze the UEP and URT properties of these codes.
For the UEP property, which means different parts of input symbols would be decoded with different error rates as there are same parts of output symbols are received. The URT property means that different parts of input symbols can be decoded with the same error rate as there are different parts of output symbols are received.
In [3], the authors interpret the URT as the UEP. As the encoding scheme which proposed in this paper, the input symbols in different blocks have different chosen probabilities when encoding an output symbol, the chosen probabilities for input symbols in each block are different, where the MIB symbols have higher chance to be selected to generate an output symbol than in LIB.
Consider a given LT code with parameters Ω(x), k, γ, where the k input symbols can be divided into a series of blocks b1, b2, . . . , bi, . . ., and the number of input symbols in each block bi is αik, where
iαi = 1. As the encoding scheme proposed in [3], input symbols in each block have their own selected probability when generating an output symbol, for the ith block, the selected probability is qi, then the input degree distribution of blocki which denotes by Λi(x) =
dΛi,dxd can be calculated as
Λi,d= ( ¯ddie−d¯i)/d!, (1) where ¯di = γqi
ddΩd
αi . The input degree distribution of block ican be rewrit- ten as
Λi(x) =ed¯iγ(x−1), (2)
and the input edge distribution can be given as λi(x) =ed¯iγ(x−1)=e
γ2qi d dΩd αi (x−1)
. (3)
For this encoding scheme, the output degree distributions of each block are allΩ(x). Therefore, we have the edge distributionω(x) =
dωdxd, whereωd= dΩd. The error rate of input symbols in each block can be calculated by
yl,i=λi
1−
d
ωd i
qi(1−yl−1,i)d−1
. (4)
As the output edge distributions are uniform, the error rate of input symbols in each block depends on their input edge distributions. As x in (3) is the prob- ability of the output symbols which could transmit “1” to their neighbors, we have 0≤x≤1. All the parameters except qi are constant, so the value ofλi(x) monotonically decreases asqi increases. Thus for theith block, the error rate of input symbols is lower asqiis larger, whatever the value of overheadγ. In other words, for a given error rate requirement, the input symbols in this block can be recovered with a lower overhead than the others. Therefore, in these codes, the UEP property can be interpret as URT.
3 Equal Recovery Time UEP Rateless Codes
In this section, we describe the proposed UEP rateless codes which provides the equal recovery time property.
The cause of UEP property can be shown by And-Or Tree analysis. For a given LT code is encoded uniformly at random, when the decoding process is finished, the probability of output symbols which could transmit “1” to its neighbor is pl. Then for an input symbol with degree d, as there ared output neighbors, the error rate of this input symbol can be calculated ased= (1−pl)d. It is not hard to find that, as 0< pl<1,ed is monotonically decreases as d increases. Hence, the input symbols with higher average degree have better error performance than the others.
Then we consider the recovery time of each input symbol for the given LT code, as the definition of the BP decoding process for LT codes. Each input sym- bol can be recovered only if it is a neighbor of an output symbol with degree 1.
Consider a moment in which the error rate of the input symbols isel, then for an output symbol with degreed, the probability this output symbol could recover one of its neighbors is (1−el)d, as this probability is monotonically decreases as dincreases, we could give the following hypothesis: The input symbol connected with output symbols with lower degrees have a higher chance to be recovered earlier.