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For IoTs of smart city scenarios always with the low cost, low power consumption, and high transmission delay properties, the traditional protocols based on feedback messages, e.g., the Automatic Repeat reQuest (ARQ) schemes, would dramatically affect the transmission efficiency. Therefore, the LT codes with only one feedback message in each entire coding process can be used to substitute the traditional protocols. As in many IoTs of smart city scenarios, the data must have both high transmission efficiency and timeliness requirements; thus, the negative effect of only the feedback message in each entire coding process cannot be neglected in such transmission environments. To enhance the transmission efficiency of such ensembles, a novel LT scheme without feedback messages is proposed in this paper. By presenting the definitions of optimal decoding overhead and recovery ratio per symbol, the optimal decoding overhead of LT codes can be found directly, then the encoding overhead of the encoder can be predesigned also. For this reason, the feedback messages in LT schemes can be removed. By using the proposed LT scheme, the transmission efficiency of IoT of smart city scenarios can be enhanced.

Being one of the scenarios of massive Internet of Things (IoT), smart cities have received a lot of attention. The IoTs of a smart city always have features such as low cost, low power consumption, small data, and a large amount of nodes [

The Low Power Wide Area Network (LPWAN) is one of the most popular access technologies of the IoTs of a smart city, which can extend the access area into tens of kilometers, and in which the wide area is provided at the price of low power, small data, high delay, etc. Some well-known IoT networks such as Lora, Narrow Band IoT (NBIoT), and SIGFOX are all included in LPWAN [

Recently, most of the researches on LPWAN are focused on access technologies, especially on the Nonorthogonal Multiple Access (NOMA), as the NOMA technology can provide a much higher amount of users access into IoTs [

The feature of LPWAN makes the traditional error control codes hard to use to guarantee the reliability of the data in the IoT of a smart city. In fact, in most of the LPWAN networks, the reliability of the data is protected by using the retransmission mechanisms. As the high delay feature exists, the transmission efficiency of these data is hard to guarantee [

Rateless codes are a class of error-control codes with an unfixed code rate. In other words, the code rate can be practically as large as needed. LT codes [

For a LT code, the decoder can send a feedback message to the encoder as there are enough output symbols (encoded packets) that have been collected, and the encoding process would be terminated as the message is received. Compared with the traditional protocols which are based on ARQ schemes, the LT codes have much fewer feedback messages. As feedback messages would lead to transmit delay, the LT code can provide greatly higher transmission efficiency than the compared protocols. Unfortunately, as in the smart city scenarios, transmission delay is large; even if the feedback channels do not exist, the only feedback message of each entire LT coding process would also impact the transmission efficiencies [

Aiming to provide higher transmission efficiency, and overcome the drawbacks of higher transmission delay and limited energy consumption, we propose a novel LT scheme without feedback messages. As the channel states (packet loss probability) can be estimated [

The paper is organized as follows. In Section 2, we review the LT codes, and the asymptotic analysis tool termed And-Or Tree analysis is also given [

In this section, we will briefly review the LT codes, including the encoding and decoding processes, and the well-known And-Or Tree analysis will also be reviewed.

First, a brief review of LT codes will be provided in the following. LT codes were proposed by Luby, which have been named as the Luby Transform (LT) codes [

LT codes are a class of erasure correct codes, which means the encoding and decoding processes of LT codes are faced to not only the bits but also the packets. For simplicity, in this paper, LT codes are defined to face the symbols; each symbol could be a bit or a packet. The original data is structured in a series of input symbols, and the encoded data is a series of output symbols. In the encoding process of LT codes, the degree distribution of output symbols are necessary, which is defined as

The most common decoding algorithm of LT codes is the Belief Propagation (BP) algorithm. The BP decoding process can be illustrated as follows: At the beginning, the decoder selects an output symbol with degree 1, then the neighbor of this output symbol can decoded. By summing the decoded input symbol on its other neighbors, the degrees of all these neighbors are minus 1, and the selected output symbol and its neighbor are moved out of the decoding process. The BP decoding process can be repeated until there is no output symbol with degree 1 that exists or there are enough input symbols decoded; the decoding processes are called as fail if there is no output symbol with degree 1 that can be found. On the other hand, the decoding processes are called as success.

The And-Or Tree analysis is a well-known technique that can be used to evaluate the performance of rateless codes [

Consider a LT code with output degree distribution

For the purpose of starting the iterative process, the expression

As

For each LT code, as the amount of the input symbols is always less than that of the collected output symbols, there are some output symbols that can be considered as

In the BP decoding process of a LT code, an output symbol with degree

In a given LT code, assuming there are

For an LT code, by given an overhead

For an output symbol with degree

It is obvious, for the output symbol, the unreleased probability can be given by

And the probability this symbol is repeated is

Then, the redundancy probability of this symbol is

Hence, Lemma

Consider an output symbol with initial degree

Let

The released probability of an output symbol with degree

It can be found that if the degree of the output symbol can be reduced to “1” or “0,” this symbol is released. Thus, the released probability of this symbol can be computed as

Then, equation (

For the traditional error control codes, the transmission efficiency can be illustrated by using the code rate, and the higher code rate would lead to a higher transmission efficiency. For LT codes, which are the first class of rateless codes, the definition of the code rate does not exist, instead of overhead. The encoder of a LT code can generate output symbols as much as is needed, then the code rate which is defined as a fixed value cannot be given. Different with the code rate, the overhead is defined as a dynamic ratio but not a fixed value. For this reason, the

To quantize the transmission efficiency of LT codes, a definition named as

To use the Monte Carlo method, it is easy to find out that there is a maximum value of RRPS, but which is helpless to quantize the transmission efficiency of a LT code. Following with the analysis method in Section 3, equation (

The RRPS of a given LT code can be represented by the ratio between the number of unredundant output symbols and the total number of output symbols and which can be given by

RRPS is the ratio between the number of unredundant output symbols and the total number of output symbols, which is defined as

As

As Lemma

For a given LT code, the RRPS is the only maximum value in the domain of definition of

By observing equation (

Let

As there is an extreme value of RRPS that exists, let

Thus, Lemma

As the RRPS directly represents the transmission efficiency of a LT code, the

In earlier studies of LT codes, an encoding process of LT codes can be terminated only if the feedback message is received. As a feedback message would be transmitted to return to the encoder until the symbol error rates are low enough, the encoding process cannot be finished at the moment that the symbol error rate just reaches the low enough level. Actually, the encoder would continue generating output symbols until the feedback message is received. In other words, the overheads of traditional LT codes would be larger than those which are needed to make sure the symbol error rates are low enough; for the scenarios with high transmission delays, the gaps between the practice overheads and needed overheads would be much larger. Hence, transmission efficiency of traditional LT codes cannot be guaranteed in IoTs of smart city scenarios.

For traditional LT codes, the symbol error rate can be determined by overhead, which means the symbol error rates would be continually decreasing with overhead growth. Although the symbol error rates of LT codes can reach an arbitrary low level if the overheads are large enough, the transmission efficiency of such a scheme would be dramatically affected. Actually, by finding out the optimal decoding overhead of a LT code, for the different reliability requirements, the symbol error rate could be guaranteed by adjusting the other parameters except the overhead. For this reason, in the proposed LT scheme without feedback messages, the various symbol error rates would be provided by using different output degree distributions.

As the different symbol error rates of different portions of LT codes can be provided by using different selection probabilities in the encoding processes [

Assuming the output degree distribution of a LT code is optimal, then the symbol error rate performance of this code would be increased as the average output symbol degree grows.

Consider LT codes with a decoding symbol error rate

It is easy to see that the error rate of this symbol would decrease as degree

Generalized to all the input symbols, it is not hard to find that the symbol error rate of an input symbol would be lower as it has been selected more times in the encoding process. In other words, symbol error rates of LT codes would increase with the average output symbol degree growing.

In this section, the symbol error rate and recovery ratio per symbol (RRPS) performances of LT codes are presented. By illustrating the symbol error rate and RRPS of the same codes, the transmission efficiencies and optimal decoding overheads can be observed. Then, we compared the proposed optimal decoding overhead with other notations which are designed to measure the corresponding decoding overhead, and the comparison of transmission efficiencies among the proposed and other related schemes in IoT smarty city scenarios are given also.

Firstly, the symbol error rate and RRPS performances of a given LT code are provided in Figure

Symbol error rate and RRPS performances of the given LT code.

Overhead

Overhead

Figure

Beside the classical output degree distribution, we also compared the symbol error rate and RRPS performances of LT codes with the robust degree distributions [

Symbol error rate performance of the LT codes with robust degree distributions.

The overall RRPS performance of the LT and LT-based codes.

We also use the RRPS to illustrate the transmission efficiency of some other LT-based codes and compared with the LT codes which are shown in Figure

Asymptotic error performance of the given LT code.

In the IoT of smart city scenarios, the small size of the data makes the traditional error correcting codes unable to provide good performances in such short length conditions; retransmission schemes based on the Automatic Repeat reQuest (ARQ) mechanism are widely adopted to provide the reliability transmission in the IoT and smart city scenarios. Because of that, the data size on each node in such scenarios is very limited; the transmission delay schemes would also lead to dramatic effects on the transmission efficiency of IoT and smart city scenarios. As the optimal decoding overhead

The transmission efficiencies of proposed and traditional LT schemes versus various target symbol error rates.

Target SER | 10^{-1} | 10^{-2} | 10^{-3} | 10^{-4} | 10^{-5} |
---|---|---|---|---|---|

Scheme 1 | 93.75% | 94.29% | 91.74% | 91.74% | 90.91% |

Scheme 2 | 85.23% | 85.72% | 83.40% | 83.40% | 82.65% |

Scheme 3 | 81.82% | 87.38% | 76.39% | 54.11% | 46.38% |

By focusing on the nonnegligible negative effect of transmission delay in IoTs of a smart city, we proposed a novel LT scheme without feedback messages in this paper. By presenting the definitions of the recovery ratio per symbol and optimal decoding overhead, and given the necessary proof, the optimal decoding overheads of LT codes can be found. By using the channel estimate techniques in the smart city scenarios, the optimal encoding overhead can be obtained, then the feedback messages of traditional LT schemes can be removed. By utilising the proposed scheme, the transmission efficiency of IoTs of a smart city can be enhanced. We compared the symbol error rate, recovery rate, and recovery ratio per symbol performances of two given LT codes. The results showed that one can easily find out the optimal decoding overhead by using the recovery ratio per symbol, and the optimal encoding overhead can be found out as well. In a word, by using the proposed LT scheme, the nonnegligible negative effect of feedback delay in IoTs can be eliminated; thus, the proposed LT scheme can dramatically improve the transmission efficiency in the IoT of smart city scenarios.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.

Shuang Wu and Qingyang Guan contributed equally to this work.

This work was supported by the Scientific Research Initiation Funds for the Doctoral Program of Xi’an International University (Grant Nos. XAIU2018070102 and XAIU2019002), Regional Innovation Capability Guidance Project (Grant No. 2021QFY01-08), General Project of the Science and Technology Department of Shaanxi Province (Grant No. 2020JM-638), and Research Foundation of Education Bureau of Hunan Province (Grant No. 20B460).