Qasim M. Chaudhari and Dr. Erchin Serpedin Department of Electrical and Computer Engineering

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.

2. Outline • Wireless sensor networks • Related work • Clock model • A Sender-Receiver protocol • Clock offset estimation • Clock offset and skew estimation • Simplified schemes • Best Linear Unbiased Estimation – Order Statistics • Minimum Variance Unbiased Estimation • Minimum Mean Square Error estimation

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

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

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

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

7. Main Challenges

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

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

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

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]

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

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

14. A Sender-Receiver Protocol Sources of error (time uncertainty) associated with message exchanges • Send time: time spent to construct a message • Access time: delays at MAC layer before actual transmission • Propagation time: time of flight from one node to another • Receive time: time needed for the receiver to receive the message and process it 2. Node B sends an ACK (Level of Node B, T1, T2, and T3) to Node A at T3. With this, Node A calculates the clock offset. 1. Node A sends a timing message (Level of Node A and T1) to Node B at T1.

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]?

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

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

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]

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

20. Clock Offset and Skew

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

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

23. Clock Offset and SkewGaussian • 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

24. Clock Offset and Skew Exponential • The likelihood function in this case is • Four different cases need to be considered

25. Clock Offset and SkewExponential Case I: known, known • Constraints • ML estimator

26. Clock Offset and SkewExponential

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

28. Clock Offset and SkewExponential • 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

29. Clock Offset and SkewExponential

30. Clock Offset and SkewExponential

31. Clock Offset and SkewExponential Case III: unknown, known • Constraints

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

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

34. Clock Offset and SkewExponential • Problem can be solved by the application of four lemmas • Final form of the ML estimator is

35. Clock Offset and SkewExponential

36. Clock Offset and SkewExponential

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

38. Utilizing Data Samples 1,N • Better skew estimation for large synchronization period • Utilize only 1st 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

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”

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

41. Utilizing Data Samples 1,N • Simulation results

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

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

44. Two Minimum Order Statistics

45. Two Minimum Order Statistics • Simulations results

46. Two Minimum Order Statistics • Computational complexity comparison with the MLE

47. Summary

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?

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

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