This paper studies the secure and accurate clock synchronization problem for sensor networks with time-varying delays and malicious attacks.A novel clock synchronization scheme based on the attack detection mechanism,...This paper studies the secure and accurate clock synchronization problem for sensor networks with time-varying delays and malicious attacks.A novel clock synchronization scheme based on the attack detection mechanism,attack compensation,and maximum consensus protocol is proposed.The proposed scheme starts with the detection of the malicious attacks and the clock data under attacks is eliminated.On the basis,software clock parameters are updated so that all the nodes in the network can have the same software skew and offset,so the clock synchronization can be achieved.Furthermore,it is theoretically proved that the proposed scheme can achieve the attack detection correctly,and further can guarantee a secure and accurate clock synchronization.In addition,extensive simulations are also conducted to validate the effectiveness of the proposed scheme.展开更多
Clock synchronization is one of the most fundamental and crucial network communication strategies.With the expansion of the Industrial Internet in numerous industrial applications,a new requirement for the precision,s...Clock synchronization is one of the most fundamental and crucial network communication strategies.With the expansion of the Industrial Internet in numerous industrial applications,a new requirement for the precision,security,complexity,and other features of the clock synchronization mechanism has emerged in various industrial situations.This paper presents a study of standardized clock synchronization protocols and techniques for various types of networks,and a discussion of how these protocols and techniques might be classified.Following that is a description of how certain clock synchronization protocols and technologies,such as PROFINET,Time-Sensitive Networking(TSN),and other well-known industrial networking protocols,can be applied in a number of industrial situations.This study also investigates the possible future development of clock synchronization techniques and technologies.展开更多
In this paper, successive lag synchronization (SLS) on a dynamical network with communication delay is investigated. In order to achieve SLS on the dynamical network with communication delay, we design linear feedba...In this paper, successive lag synchronization (SLS) on a dynamical network with communication delay is investigated. In order to achieve SLS on the dynamical network with communication delay, we design linear feedback control and adaptive control, respectively. By using the Lyapunov function method, we obtain some sufficient conditions for global stability of SLS. To verify these results, some numerical examples are further presented. This work may find potential applications in consensus of multi-agent systems.展开更多
Communication networks rely on time synchronization information generated by base station equipment(either the Global Navigation Satellite System receiver or rubidium atomic clock) to enable wireless networking and co...Communication networks rely on time synchronization information generated by base station equipment(either the Global Navigation Satellite System receiver or rubidium atomic clock) to enable wireless networking and communications. Meanwhile, the time synchronization among base stations depends on the Network Time Protocol. With the development of mobile communication systems, the corresponding time synchronization accuracy has increased as well. In this case, the use of sparsely distributed-high-precision synchronization points to synchronize time for an entire network with high precision is a key problem and is the foundation of the enhanced network communication. The current receiver equipment for China's digital synchronous network typically includes dedicated multi-channel GPS receivers for communication; however, with the development of GPS by the USA, network security has been destabilized and reliability is low. Nonetheless, network time synchronization based on Beidou satellite navigation system timing devices is an inevitable development trend for China's digital communications network with the establishment of the independently developed BDS, especially the implementation and improvement of the Beidou foundation enhancement system.展开更多
New synchronization algorithm and analysis of its convergence rate for clock oscillators in dynamical network with time-delays are presented.A network of nodes equipped with hardware clock oscillators with bounded dri...New synchronization algorithm and analysis of its convergence rate for clock oscillators in dynamical network with time-delays are presented.A network of nodes equipped with hardware clock oscillators with bounded drift is considered.Firstly,a dynamic synchronization algorithm based on consensus control strategy,namely fast averaging synchronization algorithm (FASA),is presented to find the solutions to the synchronization problem.By FASA,each node computes the logical clock value based on its value of hardware clock and message exchange.The goal is to synchronize all the nodes' logical clocks as closely as possible.Secondly,the convergence rate of FASA is analyzed that proves it is related to the bound by a nondecreasing function of the uncertainty in message delay and network parameters.Then,FASA's convergence rate is proven by means of the robust optimal design.Meanwhile,several practical applications for FASA,especially the application to inverse global positioning system (IGPS) base station network are discussed.Finally,numerical simulation results demonstrate the correctness and efficiency of the proposed FASA.Compared FASA with traditional clock synchronization algorithms (CSAs),the convergence rate of the proposed algorithm converges faster than that of the CSAs evidently.展开更多
In laboratory environment, the channel apparatus will generate particular dominant quantum noise. The noise then will give rise to some errors during synchronization. In this work, the accuracies of one qubit transpor...In laboratory environment, the channel apparatus will generate particular dominant quantum noise. The noise then will give rise to some errors during synchronization. In this work, the accuracies of one qubit transport protocol and entangled states transport protocol in the presence of noise have been studied. With the help of three important and familiar noise models, the quantum noise will degrade the accuracy has been proved. Due to the influence of quantum noise, the accuracy of entangled qubits decrease faster than that of one qubit. The entangled states will improve the accuracy in noise-free channel, and will degrade the accuracy in noise channel.展开更多
This paper proposes a distributed second-order consensus time synchronization, which incorporates the second-order consensus algorithm into wireless sensor networks. Since local clocks may have different skews and off...This paper proposes a distributed second-order consensus time synchronization, which incorporates the second-order consensus algorithm into wireless sensor networks. Since local clocks may have different skews and offsets, the algorithm is designed to include offset compensation and skew compensation. The local clocks are not directly modified, thus the virtual clocks are built according to the local clocks via the compensation parameters. Each node achieves a virtual consensus clock by periodically updated compensation parameters. Finally, the effectiveness of the proposed algorithm is verified through a number of simulations in a mesh network. It is proved that the proposed algorithm has the advantage of being distributed, asymptotic convergence, and robust to new node joining.展开更多
The data acquisition stations and the data processing center of the Science and Application Center for Lunar and Deep-space Exploration (SACLuDE) are located at different geographical sites. They respectively have the...The data acquisition stations and the data processing center of the Science and Application Center for Lunar and Deep-space Exploration (SACLuDE) are located at different geographical sites. They respectively have their own local networks and interconnect with each other through access to the core data network. This paper describes the clock drift in the computer and other networked devices building up the infrastructure of the above local networks. The network time variance of the stochastic model is also estimated. The poor precision of network synchronization will bring about potential hazards to the network operation and application running in the networks, which is clarified in the present paper. At the end of the paper, a cost-effective and feasible solution is proposed based on the Global Position System (GPS) and the Network Time Protocol (NTP).展开更多
The networked synchronization problem of a class of master-slave chaotic systems with time-varying communication topologies is investigated in this paper. Based on algebraic graph theory and matrix theory, a simple li...The networked synchronization problem of a class of master-slave chaotic systems with time-varying communication topologies is investigated in this paper. Based on algebraic graph theory and matrix theory, a simple linear state feedback controller is designed to synchronize the master chaotic system and the slave chaotic systems with a time- varying communication topology connection. The exponential stability of the closed-loop networked synchronization error system is guaranteed by applying Lyapunov stability theory. The derived novel criteria are in the form of linear matrix inequalities (LMIs), which are easy to examine and tremendously reduce the computation burden from the feedback matrices. This paper provides an alternative networked secure communication scheme which can be extended conveniently. An illustrative example is given to demonstrate the effectiveness of the proposed networked synchronization method.展开更多
Efficient multi-machine cooperation and network dynamics still remain open that jeopardize great applications in largescale machine-to-machine(M2M) networks. Among all possible machine cooperation controls, to synchro...Efficient multi-machine cooperation and network dynamics still remain open that jeopardize great applications in largescale machine-to-machine(M2M) networks. Among all possible machine cooperation controls, to synchronize tremendous machines in a timing-efficient brings one of the greatest challenge and serves as the foundation for any other network control policies. In this paper, we propose a linear-time synchronization protocol in large M2M networks. Specifically, a closed-form of synchronization rate is provided by developing the statistical bounds of the second smallest eigenvalue of the graph Laplacian matrix. These bounds enable the efficient control of network dynamics, facilitating the timing synchronization in networks. Through a practical study in Metropolis, simulation results confirm our theoretical analysis and provide effective selection of wireless technologies, including Zigbee, Wi-Fi, and cellular systems, with respect to the deployed density of machines. Therefore, this paper successfully demonstrates a practical timing synchronization, to make a breakthrough of network dynamic control in real-world machine systems, such as Internet of Things.展开更多
In this work we find a lower bound on the energy required for synchronizing moving sensor nodes in a Wireless Sensor Network (WSN) affected by large-scale fading, based on clock estimation techniques. The energy requi...In this work we find a lower bound on the energy required for synchronizing moving sensor nodes in a Wireless Sensor Network (WSN) affected by large-scale fading, based on clock estimation techniques. The energy required for synchronizing a WSN within a desired estimation error level is specified by both the transmit power and the required number of messages. In this paper we extend our previous work introducing nodes’ movement and the average message delay in the total energy, including a comprehensive analysis on how the distance between nodes impacts on the energy and synchronization quality trade-off under large-scale fading effects.展开更多
Recent advances in wireless sensor technology have enabled simultaneous exploitation of multiple channels in wireless sensor systems. In this paper, a novel time synchronization algorithm is proposed for multi- channe...Recent advances in wireless sensor technology have enabled simultaneous exploitation of multiple channels in wireless sensor systems. In this paper, a novel time synchronization algorithm is proposed for multi- channel Wireless Sensor Networks (WSNs) called Multi-Channel Time Synchronization (MCTS) protocol. Time synchronization is critical for many WSN applications and enables efficient communications between sensor nodes along with intelligent spectrum access. Contrary to many existing protocols that do not exploit multi-channel communications, the protocol takes advantage of potential multiple channels and distributes the synchronization of different nodes to distinct channels and thus, reduces the convergence time of synchronization processes significantly.展开更多
Based on a generalized chaos synchronization system and a discrete Sinai map, a non-symmetric true color (RGB) digital image secure communication scheme is proposed. The scheme first changes an ordinary RGB digital ...Based on a generalized chaos synchronization system and a discrete Sinai map, a non-symmetric true color (RGB) digital image secure communication scheme is proposed. The scheme first changes an ordinary RGB digital image with 8 bits into unrecognizable disorder codes and then transforms the disorder codes into an RGB digital image with 16 bits for transmitting. A receiver uses a non-symmetric key to verify the authentication of the received data origin, and decrypts the ciphertext. The scheme can encrypt and decz:Fpt most formatted digital RGB images recognized by computers, and recover the plaintext almost without any errors. The scheme is suitable to be applied in network image communications. The analysis of the key space, sensitivity of key parameters, and correlation of encrypted images imply that this scheme has sound security.展开更多
In order to detect the performance parameters of the network, for example, the network delay or delay jitter, the clock synchronization relations between the two hosts at two ends along the network must be calculated ...In order to detect the performance parameters of the network, for example, the network delay or delay jitter, the clock synchronization relations between the two hosts at two ends along the network must be calculated in advance. Then with the correct temporal relations between the two hosts, multimedia transmission along the network and display can occur by the proper order. A refined method based on Paxson's algorithm is proposed and testified. More accurate results can be attained by the method. By the way, the method can be used in a more complicated environment. Furthermore, an end-to-end network performance tester based on the proposed algorithm is designed and implemented.展开更多
Two-way packet exchange synchronization scheme has been widely used in wireless sensor networks. However, due to the fact that its synchronization error accumulates rapidly over hop count, its applications are greatly...Two-way packet exchange synchronization scheme has been widely used in wireless sensor networks. However, due to the fact that its synchronization error accumulates rapidly over hop count, its applications are greatly restricted. In this paper, the factors that cause the accumulation of synchronization error over hop count are investigated. Theoretical analysis shows that two factors including the clock drift and the asymmetry of two-way packet exchange, have distinct influences on synchronization error between two adjacent nodes. Further, the clock frequency order along synchronization path is found to be vital to the accumulation of synchronization error. The above three factors jointly determine the accumulation of synchronization error over hop count in wireless network. Theoretic results are also verified by three fine-grained experiments on wireless sensor network testbed. The conclusions can be used to decrease synchronization error for large-scale wireless network by careful network deployment.展开更多
In this work, the existing trade-off between time synchronization quality and energy is studied for both large-scale and small-scale fading wireless channels. We analyze the clock offset estimation problem using one-w...In this work, the existing trade-off between time synchronization quality and energy is studied for both large-scale and small-scale fading wireless channels. We analyze the clock offset estimation problem using one-way, two-way and N-way message exchange mechanisms affected by Gaussian and exponentially distributed impairments. Our main contribution is a general relationship between the total energy required for synchronizing a wireless sensor network and the clock offset estimation error by means of the transmit power, number of transmitted messages and average message delay, deriving the energy optimal lower bound as a function of the time synchronization quality and the number of hops in a multi-hop network.展开更多
基金supported by the National Natural Science Foundation of China under Grant No.61973082Six Talent Peaks Project in Jiangsu Province under Grant No.XYDXX-005。
文摘This paper studies the secure and accurate clock synchronization problem for sensor networks with time-varying delays and malicious attacks.A novel clock synchronization scheme based on the attack detection mechanism,attack compensation,and maximum consensus protocol is proposed.The proposed scheme starts with the detection of the malicious attacks and the clock data under attacks is eliminated.On the basis,software clock parameters are updated so that all the nodes in the network can have the same software skew and offset,so the clock synchronization can be achieved.Furthermore,it is theoretically proved that the proposed scheme can achieve the attack detection correctly,and further can guarantee a secure and accurate clock synchronization.In addition,extensive simulations are also conducted to validate the effectiveness of the proposed scheme.
基金supported in part by the National Key Research and Development Program of China under Grant No.2021YFB 2900100.
文摘Clock synchronization is one of the most fundamental and crucial network communication strategies.With the expansion of the Industrial Internet in numerous industrial applications,a new requirement for the precision,security,complexity,and other features of the clock synchronization mechanism has emerged in various industrial situations.This paper presents a study of standardized clock synchronization protocols and techniques for various types of networks,and a discussion of how these protocols and techniques might be classified.Following that is a description of how certain clock synchronization protocols and technologies,such as PROFINET,Time-Sensitive Networking(TSN),and other well-known industrial networking protocols,can be applied in a number of industrial situations.This study also investigates the possible future development of clock synchronization techniques and technologies.
基金Project supported by the National Natural Science Foundation of China(Grant No.61004101)the Natural Science Foundation Program of Guangxi Province,China(Grant No.2015GXNSFBB139002)+1 种基金the Graduate Innovation Project of Guilin University of Electronic Technology,China(Grant No.GDYCSZ201472)the Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation,Guilin University of Electronic Technology,China
文摘In this paper, successive lag synchronization (SLS) on a dynamical network with communication delay is investigated. In order to achieve SLS on the dynamical network with communication delay, we design linear feedback control and adaptive control, respectively. By using the Lyapunov function method, we obtain some sufficient conditions for global stability of SLS. To verify these results, some numerical examples are further presented. This work may find potential applications in consensus of multi-agent systems.
文摘Communication networks rely on time synchronization information generated by base station equipment(either the Global Navigation Satellite System receiver or rubidium atomic clock) to enable wireless networking and communications. Meanwhile, the time synchronization among base stations depends on the Network Time Protocol. With the development of mobile communication systems, the corresponding time synchronization accuracy has increased as well. In this case, the use of sparsely distributed-high-precision synchronization points to synchronize time for an entire network with high precision is a key problem and is the foundation of the enhanced network communication. The current receiver equipment for China's digital synchronous network typically includes dedicated multi-channel GPS receivers for communication; however, with the development of GPS by the USA, network security has been destabilized and reliability is low. Nonetheless, network time synchronization based on Beidou satellite navigation system timing devices is an inevitable development trend for China's digital communications network with the establishment of the independently developed BDS, especially the implementation and improvement of the Beidou foundation enhancement system.
基金Sponsored by the Cooperation Building Foundation Project of Beijing Education Committee (100070
文摘New synchronization algorithm and analysis of its convergence rate for clock oscillators in dynamical network with time-delays are presented.A network of nodes equipped with hardware clock oscillators with bounded drift is considered.Firstly,a dynamic synchronization algorithm based on consensus control strategy,namely fast averaging synchronization algorithm (FASA),is presented to find the solutions to the synchronization problem.By FASA,each node computes the logical clock value based on its value of hardware clock and message exchange.The goal is to synchronize all the nodes' logical clocks as closely as possible.Secondly,the convergence rate of FASA is analyzed that proves it is related to the bound by a nondecreasing function of the uncertainty in message delay and network parameters.Then,FASA's convergence rate is proven by means of the robust optimal design.Meanwhile,several practical applications for FASA,especially the application to inverse global positioning system (IGPS) base station network are discussed.Finally,numerical simulation results demonstrate the correctness and efficiency of the proposed FASA.Compared FASA with traditional clock synchronization algorithms (CSAs),the convergence rate of the proposed algorithm converges faster than that of the CSAs evidently.
基金Supported by The National Natural Science Foundation of China under Grant No.61075014the Science Foundation of Xi'an University of Posts and Telecommunications for Young Teachers(ZL2010-11)the Science Foundation of Shaanxi Provincial Department of Education under Grant No.11JK1051
文摘In laboratory environment, the channel apparatus will generate particular dominant quantum noise. The noise then will give rise to some errors during synchronization. In this work, the accuracies of one qubit transport protocol and entangled states transport protocol in the presence of noise have been studied. With the help of three important and familiar noise models, the quantum noise will degrade the accuracy has been proved. Due to the influence of quantum noise, the accuracy of entangled qubits decrease faster than that of one qubit. The entangled states will improve the accuracy in noise-free channel, and will degrade the accuracy in noise channel.
基金Supported by the National Natural Science Foundation of China(No.61340034)the Research Program of Application Foundation and Advanced Technology of Tianjin(No.13JCYBJC15600)
文摘This paper proposes a distributed second-order consensus time synchronization, which incorporates the second-order consensus algorithm into wireless sensor networks. Since local clocks may have different skews and offsets, the algorithm is designed to include offset compensation and skew compensation. The local clocks are not directly modified, thus the virtual clocks are built according to the local clocks via the compensation parameters. Each node achieves a virtual consensus clock by periodically updated compensation parameters. Finally, the effectiveness of the proposed algorithm is verified through a number of simulations in a mesh network. It is proved that the proposed algorithm has the advantage of being distributed, asymptotic convergence, and robust to new node joining.
文摘The data acquisition stations and the data processing center of the Science and Application Center for Lunar and Deep-space Exploration (SACLuDE) are located at different geographical sites. They respectively have their own local networks and interconnect with each other through access to the core data network. This paper describes the clock drift in the computer and other networked devices building up the infrastructure of the above local networks. The network time variance of the stochastic model is also estimated. The poor precision of network synchronization will bring about potential hazards to the network operation and application running in the networks, which is clarified in the present paper. At the end of the paper, a cost-effective and feasible solution is proposed based on the Global Position System (GPS) and the Network Time Protocol (NTP).
基金supported by the National Natural Science Foundation of China (Grant Nos. 60904046, 60972164, 60974071, and 60804006)the Special Fund for Basic Scientific Research of Central Colleges, Northeastern University, China (Grant No. 090604005)+2 种基金the Science and Technology Program of Shenyang (Grant No. F11-264-1-70)the Program for Liaoning Excellent Talents in University (Grant No. LJQ2011137)the Program for Liaoning Innovative Research Team in University (Grant No. LT2011019)
文摘The networked synchronization problem of a class of master-slave chaotic systems with time-varying communication topologies is investigated in this paper. Based on algebraic graph theory and matrix theory, a simple linear state feedback controller is designed to synchronize the master chaotic system and the slave chaotic systems with a time- varying communication topology connection. The exponential stability of the closed-loop networked synchronization error system is guaranteed by applying Lyapunov stability theory. The derived novel criteria are in the form of linear matrix inequalities (LMIs), which are easy to examine and tremendously reduce the computation burden from the feedback matrices. This paper provides an alternative networked secure communication scheme which can be extended conveniently. An illustrative example is given to demonstrate the effectiveness of the proposed networked synchronization method.
基金supported by the Major Research plan of the National Natural Science Foundation of China 9118008National Key Technology R&D Program of the Ministry of Science and Technology 2014BAC16B01
文摘Efficient multi-machine cooperation and network dynamics still remain open that jeopardize great applications in largescale machine-to-machine(M2M) networks. Among all possible machine cooperation controls, to synchronize tremendous machines in a timing-efficient brings one of the greatest challenge and serves as the foundation for any other network control policies. In this paper, we propose a linear-time synchronization protocol in large M2M networks. Specifically, a closed-form of synchronization rate is provided by developing the statistical bounds of the second smallest eigenvalue of the graph Laplacian matrix. These bounds enable the efficient control of network dynamics, facilitating the timing synchronization in networks. Through a practical study in Metropolis, simulation results confirm our theoretical analysis and provide effective selection of wireless technologies, including Zigbee, Wi-Fi, and cellular systems, with respect to the deployed density of machines. Therefore, this paper successfully demonstrates a practical timing synchronization, to make a breakthrough of network dynamic control in real-world machine systems, such as Internet of Things.
文摘In this work we find a lower bound on the energy required for synchronizing moving sensor nodes in a Wireless Sensor Network (WSN) affected by large-scale fading, based on clock estimation techniques. The energy required for synchronizing a WSN within a desired estimation error level is specified by both the transmit power and the required number of messages. In this paper we extend our previous work introducing nodes’ movement and the average message delay in the total energy, including a comprehensive analysis on how the distance between nodes impacts on the energy and synchronization quality trade-off under large-scale fading effects.
基金supported in part by TEKES(Finnish Funding Agency for Technology and Innovation)as part of the Wireless Sensor and Actuator Networks for Measurement and Control(WiSA-II)programby the U.S.Army Research Office under Cooperative Agreement W911NF-04-2-0054.
文摘Recent advances in wireless sensor technology have enabled simultaneous exploitation of multiple channels in wireless sensor systems. In this paper, a novel time synchronization algorithm is proposed for multi- channel Wireless Sensor Networks (WSNs) called Multi-Channel Time Synchronization (MCTS) protocol. Time synchronization is critical for many WSN applications and enables efficient communications between sensor nodes along with intelligent spectrum access. Contrary to many existing protocols that do not exploit multi-channel communications, the protocol takes advantage of potential multiple channels and distributes the synchronization of different nodes to distinct channels and thus, reduces the convergence time of synchronization processes significantly.
基金the National Natural Science Foundation of China under,the Foundation for University Key Teachers,高等学校博士学科点专项科研项目,教育部科学技术研究项目
文摘Based on a generalized chaos synchronization system and a discrete Sinai map, a non-symmetric true color (RGB) digital image secure communication scheme is proposed. The scheme first changes an ordinary RGB digital image with 8 bits into unrecognizable disorder codes and then transforms the disorder codes into an RGB digital image with 16 bits for transmitting. A receiver uses a non-symmetric key to verify the authentication of the received data origin, and decrypts the ciphertext. The scheme can encrypt and decz:Fpt most formatted digital RGB images recognized by computers, and recover the plaintext almost without any errors. The scheme is suitable to be applied in network image communications. The analysis of the key space, sensitivity of key parameters, and correlation of encrypted images imply that this scheme has sound security.
文摘In order to detect the performance parameters of the network, for example, the network delay or delay jitter, the clock synchronization relations between the two hosts at two ends along the network must be calculated in advance. Then with the correct temporal relations between the two hosts, multimedia transmission along the network and display can occur by the proper order. A refined method based on Paxson's algorithm is proposed and testified. More accurate results can be attained by the method. By the way, the method can be used in a more complicated environment. Furthermore, an end-to-end network performance tester based on the proposed algorithm is designed and implemented.
基金Supported by the National Natural Science Foundation of China (61003307, 61173132, 60803159),the Basic Disciplines Research Foundation of China University of Petroleum, Beijing (JCXK-2010-01), Key Laboratory of Computer System and Architecture, ICT,CAS(ICT-ARCH200901), and the Open Laboratory for the Internet Fundamental Technology, China Intemet Network Information Center(2012-N03)
文摘Two-way packet exchange synchronization scheme has been widely used in wireless sensor networks. However, due to the fact that its synchronization error accumulates rapidly over hop count, its applications are greatly restricted. In this paper, the factors that cause the accumulation of synchronization error over hop count are investigated. Theoretical analysis shows that two factors including the clock drift and the asymmetry of two-way packet exchange, have distinct influences on synchronization error between two adjacent nodes. Further, the clock frequency order along synchronization path is found to be vital to the accumulation of synchronization error. The above three factors jointly determine the accumulation of synchronization error over hop count in wireless network. Theoretic results are also verified by three fine-grained experiments on wireless sensor network testbed. The conclusions can be used to decrease synchronization error for large-scale wireless network by careful network deployment.
文摘In this work, the existing trade-off between time synchronization quality and energy is studied for both large-scale and small-scale fading wireless channels. We analyze the clock offset estimation problem using one-way, two-way and N-way message exchange mechanisms affected by Gaussian and exponentially distributed impairments. Our main contribution is a general relationship between the total energy required for synchronizing a wireless sensor network and the clock offset estimation error by means of the transmit power, number of transmitted messages and average message delay, deriving the energy optimal lower bound as a function of the time synchronization quality and the number of hops in a multi-hop network.