卷积网络编码及其应用
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摘要
信息传输的可靠性和高效性是网络传输系统的核心指标。网络编码技术是近年来提出的一种可达到网络容量上界的新技术，其核心思想是允许中间节点对输入信息做线性或非线性的编码处理后转发。相比传统路由技术，网络编码技术已经被广泛证明在提高吞吐量、数据安全、鲁棒性、普适性、负载均衡及降低计算复杂度等方面具有很大优势。特别是随机网络编码实现简单，为网络编码走向实用化提供了可能的途径。目前，网络编码已成为通信领域的研究热点。
在实际网络中，由于时延或圈的存在，不同时刻传输的消息叠加在一起，会自然产生卷积网络编码。已有的研究都是针对整个编码系统进行分析研究，在接收端译码需要知道全局编码核的全部信息，而这些信息是随着数据包一起传送到接收端的，这样译码时延比较大。对于有圈情况，全局编码核的信息是无限长的，译码不可能实现。为解决这一问题，本论文提出新颖的矩阵幂级数表示方法，从时间序列角度对卷积网络编码重新研究，给出分布式的卷积网络编码的编译码方法。其次，针对编码域大小估计困难和网络拓扑变化情况，提出一种自适应的随机卷积网络编码方法。
本论文的主要贡献包括以下几个方面。
1.保证网络编码构造成功，要求传输矩阵的行列式不为零。当网络较大时，编码域要求非常大，行列式不为零的判断是非常困难的问题。提出新颖的矩阵幂级数表示方法，利用该工具重新研究有圈网络成功构造卷积网络编码的条件，推导出卷积网络编码构造成功与否仅与本地编码核在0时刻的分量有关，这极大地简化了编码构造的复杂度。
2.已有的译码方案都是需要知道全部的全局编码核信息，这对于有圈网络上的随机网络编码是不可行的。在有限序列电路可译原理的基础上，利用矩阵幂级数工具对译码进行重新审视，给出卷积网络编码序列可译的定义，推导出编码以时延L可译的充分必要条件及推论。并提出一种分布式译码方案，可从部分全局编码核求出译码矩阵，然后逐字符译出信源消息。该方法契合了信息传输的性质，降低了译码时延及复杂度。
3.对于有圈网络，多项式的本地编码核可能会产生带分母的全局编码核，这样Erez等提出的无圈网络上的序列译码方法不再适用。我们推广了这一序列译码方法，并给出最小译码时延，其等于去掉分母后的全局编码核矩阵的行列式的最小次数与其伴随式的最大公因子的最高次数的差。
4.针对网络编码构造存在编码域大小估计困难及网络拓扑变化的问题，提出一种自适应随机卷积网络编码方法，局部地增加编码核长度直到相关的接收节点能够正确译码，解决了编码域尺寸估计困难的问题，并且随机的编码方式能够适应网络变化，提高了编码的鲁棒性，降低了译码时延。进一步分析了此算法在Combination网络和Umbrella网络以及一般网络上的增益，理论证明和仿真结果都表明该随机卷积网络编码算法很大程度降低了译码时延，且在存储需求方面有很大节省。
5.无线网络是通信领域研究的热点，如何应用网络编码技术提高无线网络的性能是一个值得研究的问题。相比有线网络，无线网络具有多播特性、噪声引起的高丢包率、信道干扰和半双工通信等特点。我们用超图建立无线网络模型，从代数角度研究无线网络编码，推导无线网络编码代数结构，给出了编码成功的代数等价条件，这为无线网络编译码算法的设计提供了强有力的理论依据。
Reliability and efficiency are two keys in modern communication systems.Network Coding is a new capacity achieving technique, where intermediate nodes areallowed to do linear or nonlinear process of the incoming information and forward. Ithas been proved to have incomparable advantages in throughput, data security,robustness, universality, load equalization and low computational complexity and so on.Meanwhile, random linear network coding (RLNC) operates simply and adapts tounknown or changing topology, which leads to possible realization. So far, networkcoding has been one of the hottest topics in Telecommunications.
In practice, due to the delay and cycles existing in networks, the messages fromdifferent times convolve together, which naturally causes convolutional network coding(CNC). The present studies on CNC are based on the whole system，and the sinks needto get all the global encoding kernels (GEKs) to decode the source messages. However,the GEKs are transmitted to sinks together mixed with messages. Therefore, it is notreasonable for a sink node waiting for receiving all the GEKs to recover the sourcemessages since large delays and high complexity are raised by the long length ofencoding kernels, infinite special for the case of cycles. To solve this problem, matrixpower series presentation is proposed to reanalyze CNC from the perspective of timesequence in this dissertation, which brings some valuable results on encoding anddecoding of CNC. Furthermore, an adaptive randomized CNC algorithm is proposed toreduce the delay and memory use.
The main contributions are listed as follows.
1. The determinant of transmission matrix should be non-zero to ensure the codesuccessful. However, it is always difficult to check as the encoding field size issufficiently large. Matrix power series representation is proposed to reformulate CNCand establish its theoretical fundamentals for practical implementations. Deduce that aCNC can be determined only by a nilpotent coefficient matrix of local encoding kernels(LEKs) at time0, which simplifies the CNC design.
2. It is not reasonable for sinks to collect all GEKs to decode the source messagesover cyclic networks, especially for random coding approach. Inspired by the theory offinite inverse circuit, a time-variant decoding model of a CNC is proposed. Newnecessary and sufficient conditions are established for the decidability of CNC at a sinknode with certain delay. They only involve the first several terms in the power series expansion of the GEK matrix. This model only deals with partial information of GEKs,and hence potentially makes CNCs applicable in a decentralized manner. This approachreduces the decoding delay and lowers the complexity.
3. For cyclic networks, polynomial fraction GEKs with non-zero denominator arepossible, and therefore the sequence decoder proposed by Erez et al. is no longerfeasible. We generalize this decoder and give the minimum decoding delay, whichequals to the difference between the minimum degree of the determinant of GEK matrixafter processing without denominators and the maximum degree of greatest commondivisor (GCD) of adjoint matrix of GEK matrix.
4. An adaptive random convolutional network coding (ARCNC) is proposed toaddress the issue of field size in RLNC for multicast, and its decoding delayperformance through both analysis and numerical simulations. ARCNC operates as aconvolutional code, with the coefficients of LEKs chosen randomly over a small finitefield. The cardinality of LEKs increases with time until the GEKs matrices at relatedsink nodes have full rank. It adapts to unknown network topologies without priorknowledge, and has reductions in decoding delay and memory overheads. We showthrough theoretic analysis and simulations that this method performs as well as RLNCin terms of decodability, and can provide significant gains in terms of average decodingdelay or memory in Combination, Umbrella, Shuttle and random geometric networks.
5. Wireless network is one hot topic in communication system, and it would beworth it to study the applications of network coding over wireless networks. Differentfrom wired networks, wireless networks have the features of broadcast, noise,interference and half-duplex and so on. To characterize the algebraic structure ofwireless network coding, a hypergragh is utilized to model wireless packet networksfrom the perspective of network layer. The algebraic description of random CNC overwireless packet networks is deduced, and the condition on coding success is alsopresented. It is shown through analysis and simulations that random CNC is capacityachieving with probability approaching1. This work provides the theoretic foundationfor wireless packet network coding.
在实际网络中，由于时延或圈的存在，不同时刻传输的消息叠加在一起，会自然产生卷积网络编码。已有的研究都是针对整个编码系统进行分析研究，在接收端译码需要知道全局编码核的全部信息，而这些信息是随着数据包一起传送到接收端的，这样译码时延比较大。对于有圈情况，全局编码核的信息是无限长的，译码不可能实现。为解决这一问题，本论文提出新颖的矩阵幂级数表示方法，从时间序列角度对卷积网络编码重新研究，给出分布式的卷积网络编码的编译码方法。其次，针对编码域大小估计困难和网络拓扑变化情况，提出一种自适应的随机卷积网络编码方法。
本论文的主要贡献包括以下几个方面。
1.保证网络编码构造成功，要求传输矩阵的行列式不为零。当网络较大时，编码域要求非常大，行列式不为零的判断是非常困难的问题。提出新颖的矩阵幂级数表示方法，利用该工具重新研究有圈网络成功构造卷积网络编码的条件，推导出卷积网络编码构造成功与否仅与本地编码核在0时刻的分量有关，这极大地简化了编码构造的复杂度。
2.已有的译码方案都是需要知道全部的全局编码核信息，这对于有圈网络上的随机网络编码是不可行的。在有限序列电路可译原理的基础上，利用矩阵幂级数工具对译码进行重新审视，给出卷积网络编码序列可译的定义，推导出编码以时延L可译的充分必要条件及推论。并提出一种分布式译码方案，可从部分全局编码核求出译码矩阵，然后逐字符译出信源消息。该方法契合了信息传输的性质，降低了译码时延及复杂度。
3.对于有圈网络，多项式的本地编码核可能会产生带分母的全局编码核，这样Erez等提出的无圈网络上的序列译码方法不再适用。我们推广了这一序列译码方法，并给出最小译码时延，其等于去掉分母后的全局编码核矩阵的行列式的最小次数与其伴随式的最大公因子的最高次数的差。
4.针对网络编码构造存在编码域大小估计困难及网络拓扑变化的问题，提出一种自适应随机卷积网络编码方法，局部地增加编码核长度直到相关的接收节点能够正确译码，解决了编码域尺寸估计困难的问题，并且随机的编码方式能够适应网络变化，提高了编码的鲁棒性，降低了译码时延。进一步分析了此算法在Combination网络和Umbrella网络以及一般网络上的增益，理论证明和仿真结果都表明该随机卷积网络编码算法很大程度降低了译码时延，且在存储需求方面有很大节省。
5.无线网络是通信领域研究的热点，如何应用网络编码技术提高无线网络的性能是一个值得研究的问题。相比有线网络，无线网络具有多播特性、噪声引起的高丢包率、信道干扰和半双工通信等特点。我们用超图建立无线网络模型，从代数角度研究无线网络编码，推导无线网络编码代数结构，给出了编码成功的代数等价条件，这为无线网络编译码算法的设计提供了强有力的理论依据。
Reliability and efficiency are two keys in modern communication systems.Network Coding is a new capacity achieving technique, where intermediate nodes areallowed to do linear or nonlinear process of the incoming information and forward. Ithas been proved to have incomparable advantages in throughput, data security,robustness, universality, load equalization and low computational complexity and so on.Meanwhile, random linear network coding (RLNC) operates simply and adapts tounknown or changing topology, which leads to possible realization. So far, networkcoding has been one of the hottest topics in Telecommunications.
In practice, due to the delay and cycles existing in networks, the messages fromdifferent times convolve together, which naturally causes convolutional network coding(CNC). The present studies on CNC are based on the whole system，and the sinks needto get all the global encoding kernels (GEKs) to decode the source messages. However,the GEKs are transmitted to sinks together mixed with messages. Therefore, it is notreasonable for a sink node waiting for receiving all the GEKs to recover the sourcemessages since large delays and high complexity are raised by the long length ofencoding kernels, infinite special for the case of cycles. To solve this problem, matrixpower series presentation is proposed to reanalyze CNC from the perspective of timesequence in this dissertation, which brings some valuable results on encoding anddecoding of CNC. Furthermore, an adaptive randomized CNC algorithm is proposed toreduce the delay and memory use.
The main contributions are listed as follows.
1. The determinant of transmission matrix should be non-zero to ensure the codesuccessful. However, it is always difficult to check as the encoding field size issufficiently large. Matrix power series representation is proposed to reformulate CNCand establish its theoretical fundamentals for practical implementations. Deduce that aCNC can be determined only by a nilpotent coefficient matrix of local encoding kernels(LEKs) at time0, which simplifies the CNC design.
2. It is not reasonable for sinks to collect all GEKs to decode the source messagesover cyclic networks, especially for random coding approach. Inspired by the theory offinite inverse circuit, a time-variant decoding model of a CNC is proposed. Newnecessary and sufficient conditions are established for the decidability of CNC at a sinknode with certain delay. They only involve the first several terms in the power series expansion of the GEK matrix. This model only deals with partial information of GEKs,and hence potentially makes CNCs applicable in a decentralized manner. This approachreduces the decoding delay and lowers the complexity.
3. For cyclic networks, polynomial fraction GEKs with non-zero denominator arepossible, and therefore the sequence decoder proposed by Erez et al. is no longerfeasible. We generalize this decoder and give the minimum decoding delay, whichequals to the difference between the minimum degree of the determinant of GEK matrixafter processing without denominators and the maximum degree of greatest commondivisor (GCD) of adjoint matrix of GEK matrix.
4. An adaptive random convolutional network coding (ARCNC) is proposed toaddress the issue of field size in RLNC for multicast, and its decoding delayperformance through both analysis and numerical simulations. ARCNC operates as aconvolutional code, with the coefficients of LEKs chosen randomly over a small finitefield. The cardinality of LEKs increases with time until the GEKs matrices at relatedsink nodes have full rank. It adapts to unknown network topologies without priorknowledge, and has reductions in decoding delay and memory overheads. We showthrough theoretic analysis and simulations that this method performs as well as RLNCin terms of decodability, and can provide significant gains in terms of average decodingdelay or memory in Combination, Umbrella, Shuttle and random geometric networks.
5. Wireless network is one hot topic in communication system, and it would beworth it to study the applications of network coding over wireless networks. Differentfrom wired networks, wireless networks have the features of broadcast, noise,interference and half-duplex and so on. To characterize the algebraic structure ofwireless network coding, a hypergragh is utilized to model wireless packet networksfrom the perspective of network layer. The algebraic description of random CNC overwireless packet networks is deduced, and the condition on coding success is alsopresented. It is shown through analysis and simulations that random CNC is capacityachieving with probability approaching1. This work provides the theoretic foundationfor wireless packet network coding.
引文
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