One of the central issues in solving differential equations by numerical methods is the issue of approximation. The standard way of approximating differential equations by numerical methods (particularly difference me...One of the central issues in solving differential equations by numerical methods is the issue of approximation. The standard way of approximating differential equations by numerical methods (particularly difference methods) is to question the degree of approximation in the form O(h<sup>p</sup>). Here h is the grid step. In this case we have an implicit approximation. Based on the difference equation approximating the differential equation, the order of approximation is obtained using the Taylor series. However, it is possible to calculate the approximation error at nodal points based on the method of moving nodes. The method of moving nodes allows obtaining an approximate analytical expression. On the basis of the approximate form, it is possible to calculate the approximation error. The analytical form of the approximation makes it possible to efficiently calculate this error. On the other hand, the property of this error allows the construction of new improved circuits. In addition, based on these types of errors, you can create a differential analog of the difference equation that gives an exact approximation.展开更多
Network Coding (NC) is a recent technique which is used to improve the transmission data rate and the power efficiency. These goals are obtained by combining data together before transmitting them, resulting to less t...Network Coding (NC) is a recent technique which is used to improve the transmission data rate and the power efficiency. These goals are obtained by combining data together before transmitting them, resulting to less transmitted data that carry the same amount of information. NC research work over the physical layer and the upper layers are popular and needed to be more investigated. In this paper, we propose a practical system of large-number of connected multi-source network coding (LMSNC), at the physical layer that exploits the broadcast nature of the wireless channel, using the practical and bandwidth-efficient schemes decode-and-forward (DF) and then compare it with Amplify and Forward (AF). The theoretical analysis and the simulation results show the effect of the noise when it cumulates in AF system and how DF is solving this severe default. Moreover, we consider the MSNC for Small-number of connected sources (SMSNC) and the two-way communication setup where two users exchange their information over an intermediate network node (ideally called Base Station), as two reference cases to compare with. With SMSNC, the number of necessary downlink transmissions from the intermediate node to the users is reduced, and thus the throughput is increased. Simulation results obtained using high-performance non-binary turbo codes, based on Partial Unit Memory (PUM) codes (4, 2, 1, 4) and (8, 4, 3, 8);confirm that combining PUM Turbo Code (PUMTC) and NC in the proposed MSNC setup gives almost the same BER performance as that for SMSNC at the small number of processing steps mainly when PUMTC (8, 4, 3, 8) is performed, which is required to retrieve the received coded messages. In the scenario of AF, combining packets results to cumulate the noise, which justifies the reason we decided to increase the number of transmitted coded messages in the network, i.e., the BER performance improves when sending extra coded messages. Finally, the possibility for a trade-off among BER, data rate and the number of transmitted coded messages is shown for LMSNC through graphics and simulation results.展开更多
文摘One of the central issues in solving differential equations by numerical methods is the issue of approximation. The standard way of approximating differential equations by numerical methods (particularly difference methods) is to question the degree of approximation in the form O(h<sup>p</sup>). Here h is the grid step. In this case we have an implicit approximation. Based on the difference equation approximating the differential equation, the order of approximation is obtained using the Taylor series. However, it is possible to calculate the approximation error at nodal points based on the method of moving nodes. The method of moving nodes allows obtaining an approximate analytical expression. On the basis of the approximate form, it is possible to calculate the approximation error. The analytical form of the approximation makes it possible to efficiently calculate this error. On the other hand, the property of this error allows the construction of new improved circuits. In addition, based on these types of errors, you can create a differential analog of the difference equation that gives an exact approximation.
文摘Network Coding (NC) is a recent technique which is used to improve the transmission data rate and the power efficiency. These goals are obtained by combining data together before transmitting them, resulting to less transmitted data that carry the same amount of information. NC research work over the physical layer and the upper layers are popular and needed to be more investigated. In this paper, we propose a practical system of large-number of connected multi-source network coding (LMSNC), at the physical layer that exploits the broadcast nature of the wireless channel, using the practical and bandwidth-efficient schemes decode-and-forward (DF) and then compare it with Amplify and Forward (AF). The theoretical analysis and the simulation results show the effect of the noise when it cumulates in AF system and how DF is solving this severe default. Moreover, we consider the MSNC for Small-number of connected sources (SMSNC) and the two-way communication setup where two users exchange their information over an intermediate network node (ideally called Base Station), as two reference cases to compare with. With SMSNC, the number of necessary downlink transmissions from the intermediate node to the users is reduced, and thus the throughput is increased. Simulation results obtained using high-performance non-binary turbo codes, based on Partial Unit Memory (PUM) codes (4, 2, 1, 4) and (8, 4, 3, 8);confirm that combining PUM Turbo Code (PUMTC) and NC in the proposed MSNC setup gives almost the same BER performance as that for SMSNC at the small number of processing steps mainly when PUMTC (8, 4, 3, 8) is performed, which is required to retrieve the received coded messages. In the scenario of AF, combining packets results to cumulate the noise, which justifies the reason we decided to increase the number of transmitted coded messages in the network, i.e., the BER performance improves when sending extra coded messages. Finally, the possibility for a trade-off among BER, data rate and the number of transmitted coded messages is shown for LMSNC through graphics and simulation results.
