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1 Estimation of Clock Parameters and Performance Benchmarks for Synchronization in Wireless Sensor Networks Qasim M. Chaudhari and Dr. Erchin Serpedin Department of Electrical and Computer Engineering Texas A&M University, College Station, TX.

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2 Outline 1. Wireless sensor networks 2. Related work 3. Clock model 4. A Sender-Receiver protocol 5. Clock offset estimation 6. Clock offset and skew estimation 7. Simplified schemes 8. Best Linear Unbiased Estimation – Order Statistics 9. Minimum Variance Unbiased Estimation 10. Minimum Mean Square Error estimation

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3 11. Clock synchronization of inactive nodes 12. Clock offset and skew estimation in a Receiver-Receiver protocol 13. Conclusions 14. Future research directions

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4 Wireless Sensor Networks Gateway Internet Server Wireless Terminal S S D Source Destination D S

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5 Introduction Small scale sensor nodes Limited power Harsh environmental conditions Communication failures Node failures Dynamic network topology Mobility of nodes

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6 Applications Monitoring Environment and habitat Military surveillance Security Traffic Controlling and tracking Industrial processes Fire Detection Object tracking

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7 Main Challenges

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8 Importance of time synchronization Time synchronization in WSNs is important for Efficient duty cycling Localization and location-based monitoring Data fusion Distributed beamforming and target tracking Security protocols Network scheduling and routing, TDMA

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9 Constraints Limited hardware Reduced computational power Low memory Limited energy Communication vs. computation RF energy required to transmit 1 bit over 100 meters is equivalent to execution of 3 million instructions [Pottie 00] Traditional clock synchronization techniques Communication comes for free Computational resources are powerful Examples: NTP is energy expensive, GPS is cost expensive

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10 Related Work Reference Broadcast Synchronization (RBS) [Elson 02] Conventional receiver-receiver protocol Reduces nondeterministic delays Conserves energy via post facto synchronization Timing synch Protocol for Sensor Networks (TPSN) [Ganeriwal 03] Conventional sender-receiver protocol Two operational phases: Level Discovery and Synchronization Time Diffusion Protocol (TDP) [Su 05] Achieves a network-wide equilibrium time using an iterative, weighted averaging technique based on diffusion of timing messages

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11 Related Work Analysis of a sender-receiver model [Ghaffar 02] For known fixed delays, maximum likelihood estimator for clock offset does not exist Five algorithms: median round delay, minimum round delay, minimum link delay, median phase, average phase. Minimum link delay algorithm has the lowest variance Maximum likelihood clock offset estimator for unknown fixed delays [Jeske 05]

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12 Clock Model A computer clock consists of two components Frequency source Means of accumulating timing events Practical clocks are set with limited precision Frequency sources run at slightly different rates Frequency of a crystal oscillator varies due to Initial manufacturing tolerance Temperature, pressure Aging

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13 Clock Model A general clock model can be represented by where is the clock offset, is the clock skew and is the clock drift Clock synchronization problem Given the logical clock for a node k in the network, then is a function of Target synchronization accuracy Amount of energy the network is willing to pay

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14 1. Node A sends a timing message (Level of Node A and T 1 ) to Node B at T 1. 2. Node B sends an ACK (Level of Node B, T 1, T 2, and T 3 ) to Node A at T 3. With this, Node A calculates the clock offset. Sources of error (time uncertainty) associated with message exchanges o Send time: time spent to construct a message o Access time: delays at MAC layer before actual transmission o Propagation time: time of flight from one node to another o Receive time: time needed for the receiver to receive the message and process it A Sender-Receiver Protocol

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15 Observations Fixed clock offset model is not sufficient in practice Clock skew correction results in long term synchronization and hence more energy savings Network delays being asymmetric is a more realistic scenario Even for the symmetric clock offset only model, better estimation schemes achieving are possible Minimum Variance Unbiased Estimation (MVUE) Minimum Mean Square Error Estimation (MMSE) Lack of analytical performance bounds and metrics Average RBS error: [Elson 02] or [Ganeriwal 03]?

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16 Clock Offset Gaussian Noise Assumption One motivation comes from experimental basis [Elson 02] In case of unknown delay distribution, we can evoke Central Limit theorem Example: for uniform delays, the sum of even two of them starts resembling a Gaussian curve

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17 Clock Offset The likelihood function can be written as And the clock offset estimate and the CRLB are

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18 Clock Offset Exponential Delay Assumption Random delays often modeled as exponential Several traces of delay measurements on Internet collected by [Moon 99] fitting an exponential distribution Conformation of experimental observations with mathematical results Experimental observations Minimum link delay algorithm [Paxson 98] Clock Filter algorithm in NTP [Mills 91] Mathematical results Best performance by Minimum link delay algorithm [Ghaffar 02] ML estimate based on minimum order statistics [Jeske 05]

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19 Clock Offset Likelihood function is given as ML clock offset estimate is CRLB is derived as

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20 Clock Offset and Skew

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21 Clock Offset and Skew Gaussian Likelihood function with is Joint ML estimate for clock offset is shown to be where

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22 Clock Offset and Skew Gaussian And for the clock skew Computationally quite complex Fixed delay must be known Open problem: Recursive implementation/update?

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23 Clock Offset and Skew Gaussian Cramer-Rao Lower Bound is expressed as where Proportional to clock skew squared Not only dependent on number of synchronization messages but also on the synchronization period

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24 Clock Offset and Skew Exponential The likelihood function in this case is Four different cases need to be considered Case IKnown Case IIKnownUnknown Case IIIUnknownKnown Case IVUnknown

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25 Clock Offset and Skew Exponential Case I: known, known Constraints ML estimator

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26 Clock Offset and Skew Exponential

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27 Clock Offset and Skew Exponential Case II: known, unknown Constraints Lemma 1: lies on one of the following curves

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28 Clock Offset and Skew Exponential Lemma 2: lies either on point A or to the left of it (B,C,…) Lemma 3: To the left of A, boundary of support region is formed by a sequence of curves with decreasing slopes Lemma 4: is unique and is given by one of

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29 Clock Offset and Skew Exponential

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30 Clock Offset and Skew Exponential

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31 Clock Offset and Skew Exponential Case III: unknown, known Constraints

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32 Clock Offset and Skew Exponential Lemma 5: Only two points satisfy the constraints ML estimator has the closed-form expression

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33 Clock Offset and Skew Exponential Case IV: unknown, unknown Constraints Curves intersect on the line Over this line, is constrained by

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34 Clock Offset and Skew Exponential Problem can be solved by the application of four lemmas Final form of the ML estimator is

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35 Clock Offset and Skew Exponential

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36 Clock Offset and Skew Exponential

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37 Simplified Schemes Fixed delay must be known in Gaussian case Computational complexity Further simplification within the same framework is possible suitable for WSNs in case Synchronization accuracy constraints are not stringent Energy conservation constraints are strict One simple scheme is independent of delay distribution involved Cost paid is slight degradation in estimation quality

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38 Utilizing Data Samples 1,N Better skew estimation for large synchronization period Utilize only 1 st and last sample differences for eliminating the clock offset Define Simplified new model where and are either Gaussian or Laplacian distributed depending on original delay distribution

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39 Utilizing Data Samples 1,N Gaussian delays Likelihood function for highly reduced data set is ML-Like clock skew estimator is expressed as CRLB-Like lower bound is Depends on timestamping “distances”

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40 Utilizing Data Samples 1,N Exponential delays The reduced likelihood function is ML-Like clock skew estimator can be derived as CRLB-Like lower bound

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41 Utilizing Data Samples 1,N Simulation results

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42 Two Minimum Order Statistics Motivation Unknown delay distribution Small synchronization period Opening the model equations as Choose two points as

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43 Two Minimum Order Statistics Joint the two points to obtain the estimate through its slope and intercept The form of the estimator is Almost as simple as the clock offset only case Knowledge of is not required

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44 Two Minimum Order Statistics

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45 Two Minimum Order Statistics Simulations results

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46 Two Minimum Order Statistics Computational complexity comparison with the MLE

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47 Summary GaussianExponential Offset ModelMLE + CRLBCRLB Offset + Skew ModelMLE + CRLBMLE + Algorithms Offset + Skew ModelUsing First and Last sample ML-Like + LB Using First and Last sample ML-Like + LB Offset + Skew ModelTwo minimum order statistics Algorithm + Computational Complexity

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48 Best Linear Unbiased Estimation – Order Statistics Limited power resources in WSN implies better estimation techniques should be utilized Results derived so far correspond to symmetric delays, although asymmetry is a more realistic scenario Best Linear Unbiased Estimation (BLUE) is suboptimal in general due to linearity constraint What if the linearity constraints are on the order statistics of observed data, instead of the raw observations?

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49 Best Linear Unbiased Estimation – Order Statistics Transforming the data as Following relations hold for ordered data

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50 Best Linear Unbiased Estimation – Order Statistics The covariance matrix for can be derived as Its inverse can be found by Gauss-Jordan elimination Let the ordered observations be represented as

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51 Best Linear Unbiased Estimation – Order Statistics Asymmetric Link Delays The asymmetric linear model can be written as And the Gauss-Markov theorem implies

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52 Best Linear Unbiased Estimation – Order Statistics The covariance matrix for is The final expression for is

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53 Best Linear Unbiased Estimation – Order Statistics Symmetric Link Delays The symmetric linear model can be written as The Gauss-Markov theorem yields the solution

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54 Best Linear Unbiased Estimation – Order Statistics Covariance for is The expression for is BLUE-OS for same as the MLE for symmetric link delays

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55 Minimum Variance Unbiased Estimation Asymmetric Link Delays Found by the application of Rao-Blackwell-Lehmann-Scheffe theorem Likelihood function can be expressed as According to Neyman-Fisher factorization theorem, the sufficient statistics is

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56 Minimum Variance Unbiased Estimation Notice that Find such that Applicable only if is a complete sufficient statistic Only function of is unbiased

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57 Minimum Variance Unbiased Estimation Unbiased estimator of ? Note that BLUE-OS is unbiased and hence MVUE ! Compensation for asymmetry through Variance of the clock offset

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58 Minimum Variance Unbiased Estimation Symmetric Link Delays Again applying the Rao-Blackwell-Lehmann-Scheffe theorem, the likelihood function is More than one unbiased functions of complete statistic? Through Neyman-Fisher factorization theorem, the actual sufficient statistics is

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59 Minimum Variance Unbiased Estimation is proved to be complete Unbiased estimator of ? BLUE-OS is unbiased and hence the MVUE In symmetric case, the MVUE and BLUE-OS of coincide with MLE Its variance is

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60 Summary Clock Offset MVUE Symmetric DelaysMSE Remarks Asymmetric DelaysMVUE MSE Remarks

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61 Explanatory Remarks Does this discontinuity in clock offset estimates performance make sense? Which estimator is better when the network delays are slightly symmetric? MVUE is the best in unbiased class of estimators, not all. For asymmetric case,

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62 Explanatory Remarks The MLE outperforms the MVUE under the condition Estimator could be chosen according to the number of synchronization messages if knowledge of is available Around the point, MLE attains lesser MSE as the link asymmetry decreases, i.e.,

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63 Explanatory Remarks Simulations results

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64 Explanatory Remarks Apparently, adapting between the two estimators a good idea according to since have been obtained too. Despite the fact that MLE is functionally invariant, considerable amplification of errors occurs due to repeated nonlinear processing Results are even applicable to Internet time synchronization

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65 Explanatory Remarks As a byproduct, the MVUE of the fixed and mean variable link delays are obtained Endd-to-end delay measurements are helpful in analyzing network performance Very useful for applications behaving adaptively based on observed network performance Continuous media applications, such as audio and video, absorb the delay jitter perceived at receiver for smooth playout of media stream

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66 Minimum Mean Square Error Estimation In general, the MMSE estimator is not realizable due to the dependence of MSE on the unknown parameter MSE is a sum of variance and bias squared This dependence usually comes from the bias Setting the MSE proportional to inverse of the scale parameter cancels the dependent factors

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67 Minimum Mean Square Error Estimation The MMSE estimator comes out to be a function of MVUE Closed-form expression for And for mean link delays MSE of clock offset

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68 Minimum Mean Square Error Estimation The MMSE estimator comes out to be a function of MVUE Closed-form expression for And for mean link delays MSE of clock offset

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69 Clock Synchronization for Inactive Nodes Packet synchronization protocols Receiver-Receiver (R-R) Sender-Receiver (S-R) For WSNs implementing any sender-receiver protocol, the inactive nodes can exploit the timing messages received

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70 Clock Synchronization for Inactive Nodes The model can be represented as

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71 Clock Synchronization for Inactive Nodes Likelihood function, assuming symmetric delays, is The maximum likelihood estimator is derived as

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72 Clock Synchronization for Inactive Nodes

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73 Clock Synchronization for Inactive Nodes The pdf of is obtained as Cramer-Rao Lower Bound is Is the ML estimator efficient?

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74 Clock Synchronization for Inactive Nodes An efficient estimator does not exist due to the rule “if an efficient estimator exists, the ML procedure will produce it”. Simulation results

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75 Clock Synchronization for Inactive Nodes Symmetric delay assumption was less realistic Better estimation techniques can be employed Using the transformed data, the linear model can be written as

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76 Clock Synchronization for Inactive Nodes and Hence, the covariance matrix is Final form of estimator

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77 Clock Synchronization for Inactive Nodes The likelihood function in symmetric case is MVUE for the clock offset is derived and shown to coincide with BLUE-OS and MLE Its variance is give by

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78 Clock Synchronization for Inactive Nodes Similarly, for asymmetric link delays, the BLUE-OS is which is also the MVUE Its variance is given by

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79 Clock Synchronization for Inactive Nodes MMSE estimator can be derived as a function of MVUE Closed-form expression for And for mean link delays MSE of clock offset

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80 Clock Offset and Skew in a Receiver-Receiver Protocol Main sources of errors – send time and channel access time – are removed A receiver-receiver model can be represented as

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81 Clock Offset and Skew in a Receiver-Receiver Protocol The likelihood function can be expressed as where Objective function to be maximized is over the constraints

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82 Clock Offset and Skew in a Receiver-Receiver Protocol JML estimator

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83 Clock Offset and Skew in a Receiver-Receiver Protocol Why Gibbs Sampler? Joint ML estimator is biased MVUE does not exist since the sufficient statistics depend on unknown parameters Posterior distribution can be found and clock parameters can be estimated by its mean for better results Straightforward extension to additional unknown parameters, e.g., clock drift Posterior distribution involves complex integrations, hence the Markov-Chain Monte Carlo (MCMC) methods

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84 Clock Offset and Skew in a Receiver-Receiver Protocol Algorithm for Gibbs Sampling is to iterate the following with initial values : After a threshold value, the set behaves as the sample values from the joint posterior

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85 Clock Offset and Skew in a Receiver-Receiver Protocol In the current scenario, the Gibbs Sampler is implemented as follows Performs better than the JML estimator Simulated with MVUE as the lower bound with one parameter known BLUE as the upper bound due to linearity constraints

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86 Clock Offset and Skew in a Receiver-Receiver Protocol Simulation results

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87 Conclusions General exponential family model Accumulated error analysis for multihop protocols Effect of mobility on time synchronization

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88 Future Research Directions General exponential family model Accumulated error analysis for multihop protocols Effect of mobility on time synchronization

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