Efficient Predictive Estimator for Holdover in GPS

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Information for Contributors............................................................................................................................................................... 2113 Multimedia Example........................................................................................................................................................... 2118 PAPERS Ferroelectrics Double Synchronized Switch Harvesting (DSSH): A New Energy Harvesting Scheme for Efficient Energy Extraction................. . ................................................................................................... M. Lallart, L. Garbuio, L. Petit, C. Richard, and D. Guyomar 2119 Frequency Control Efficient Predictive Estimator for Holdover in GPS-based Clock Synchronization....................Y. S. Shmaliy and L. Arceo-Miquel 2131 Medical Ultrasonics Two-Dimensional Blind Bayesian Deconvolution of Medical Ultrasound Images............................................. R. Jiřík and T. Taxt Pulse Inversion Sequences For Mechanically Scanned Transducers.................................................................................................. . .......................................................................................... M. E. Frijlink, D. E. Goertz, N. de Jong, and A. F. W. van der Steen Dual High-Frequency Difference Excitation for Contrast Detection......................... C.-K. Yeh, S.-Y. Su, C.-C. Shen, and M.-L. Li Assessment of the Mechanical Properties of the Musculoskeletal System Using 2-D and 3-D Very High Frame Rate Ultrasound . ............................................................................................................ T. Deffieux, J.-L. Gennisson, M. Tanter, and M. Fink  Ultrasonic Assessment of Cortical Bone Thickness In Vitro and In Vivo........................................................................................... . ........................................................................................... J. Karjalainen, O. Riekkinen, J. Töyräs, H. Kröger, and J. Jurvelin Sidelobe Suppression in Ultrasound Imaging Using Dual Apodization with Cross-Correlation..................C. H. Seo and J. T. Yen Coded Ultrasound for Blood Flow Estimation Using Subband Processing.....F. Gran, J. Udesen, M. B. Nielsen, and J. A. Jensen A Composite High-Frame-Rate System for Clinical Cardiovascular Imaging...................................................................................  . .......................................................................................... S. Wang, W.-N. Lee, J. Provost, J. Luo, and E. E. Konofagou

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Sensors, NDE, and Industrial Applications A Frame-Based Approach for Wideband Correlation-Inversion of Lossless Scatterers.................................................A. Shlivinski 2234 Viscosity Measurement of Newtonian Liquids Using the Complex Reflection Coefficient.............................................................. . ...................................................................................................... E. E. Franco, J. C. Adamowski, R. T. Higuti, and F. Buiochi 2247 Defect Characterization Using an Ultrasonic Array to Measure the Scattering Coefficient Matrix................................................... . ............................................................................................................................. J. Zhang, B. W. Drinkwater, and P. D. Wilcox 2254 Physical Acoustics Friction Drive of an SAW Motor. Part III: Modeling................................................................T. Shigematsu and M. K. Kurosawa Friction Drive of an SAW Motor. Part IV: Physics of Contact..................................................T. Shigematsu and M. K. Kurosawa Friction Drive of an SAW Motor. Part V: Design Criteria.........................................................T. Shigematsu and M. K. Kurosawa Flow Patterns and Transport in Rayleigh Surface Acoustic Wave Streaming: Combined Finite Element Method and Raytracing Numerics versus Experiments...................................T. Frommelt, D. Gogel, M. Kostur, P. Talkner, P. Hänggi, and A. Wixforth

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Iterative High-Resolution Wavenumber Inversion Applied to BroadbandAcoustic Data.................................................................. ................................................................................................................................... F. D. Philippe, P. Roux, and D. Cassereau 2306 A System of Two Piezoelectric Transducers and a Storage Circuit for Wireless Energy Transmission through a Thin Metal Wall ............................................................................................................................................. H. Hu, Y. Hu, C. Chen, and J. Wang 2312 Coupled Resonator Filter with Single-Layer Acoustic Coupler.................... T. Jamneala, M. Small, R. Ruby, and J. D. Larson III 2320 Transducers and Transducer Materials Vibration Characteristics of a Circular Cylindrical Panel Piezoelectric Transducer.........Z. Yang, J. Yang, Y. Hu, and Q.-M. Wang 2327 Characterization of Dual-Electrode CMUTs: Demonstration of Improved Receive Performance and Pulse Echo Operation with Dynamic Membrane Shaping......................................... R. O. Guldiken, M. Balantekin, J. Zahorian, and F. L. Degertekin  2336 CORRESPONDENCE Analysis and Evaluation of a Novel Quality Assurance Device for Ultrasonic Medical Imaging Systems.... Z. Hah and R. Naum 2345

Composite Electromechanical Wave Imaging at 481 Hz Images of the propagation of an electromechanical wave (in red, denoted by the white arrows) along the posterior wall of a normal human left ventricle in a longaxis view during systole are shown. The total time span of the wave propagation was 31 ms from a heart-beat total duration of 857 ms. ECG traces are not shown here, but are provided in the accompanying paper on page 2221. The white arrows indicate the wavefront propagating along the posterior wall from the apical to the basal side. These images were obtained at a frame rate of 481 Hz using a new technique, the ECG-gated composite method, for full-view, high beam density, echocardiographic and elasticity imaging. Axial displacements were calculated using 1-D cross-correlation on consecutive radio-frequency frames. Color bars represent the axial displacements in mm scale for all images shown. Displacements towards the transducer (top) were coded in red. This technique can be used as a noninvasive method for mapping the conduction and contractility properties in both normal and pathological hearts. Abnormal wave propagation patterns may thus be proven to be key for the early detection and reliable depiction of heart disease. Images courtesy of Shougang Wang, Wei-Ning Lee, Jean Provost, Jianwen Luo, and Elisa E. Konofagou. The contributors are with the Department of Biomedical Engineering, Columbia University, New York. E. E. Konofagou is also with the Department of Radiology at Columbia University. This work was supported by the National Institutes of Health (R01EB006042), the American Heart Association (SDG0435444T) and the Wallace H. Coulter Foundation (WHCFCU02650301).

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IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control ,

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Efficient Predictive Estimator for Holdover in GPS-based Clock Synchronization Yuriy S. Shmaliy, Senior Member, IEEE, and Luis Arceo-Miquel the AT-cut quartz crystal resonator has demonstrated an extremely small long-term frequency drift of about −3 × 10−10/year [11]. Based on prediction, the clock or oscillator holdover technology prolongs continued functioning time after loss of a synchronizing signal. For the clock to operate in holdover, prediction is provided of the time error at a current discrete time n by processing the nearest available past measurements. A control loop is then organized such that the synchronizing signal is produced by prediction. To organize prediction, different approaches can be used. One may employ extrapolation methods or prolong an interpolating function (e.g., Lagrange and Newton methods and I. Introduction splines) [12] and regression curve from the past to n that he problem of accuracy of wirelessly disciplined local have been applied to clocks by many authors [13]–[19]. clocks and oscillators arises whenever a synchroniz- Setting aside these methods, the 1 step toward predictive ing signal is temporary not available. The relevant opera- FIR filters was derived by Heinonen and Neuvo in [20] tion mode is termed “holdover” and technical solutions employing the Lagrange multipliers and, thereby, developfor holdover are often crucial for both ground and space ing the approach proposed by Johnson [21]. It was also applications. Holdover can also be implied for available, stated in [13] that linear predictors are optimal or close to optimal for the prediction of clock instabilities. We find but “bad,” synchronizing signals. In different applications, it is required to maintain nor- advantages of linear prediction in many papers [22]–[24]. Representation of the clock model in state space allows mal functioning in holdover during hours, days, or even months on the ground [1]–[4] and on a board [5], [6]. In solving the problem from the standpoint of system theory deep space, oscillators and clocks typically operate with- using the Kalman filter. Suitable for holdover, the predicout synchronizing signals, meaning that normal function- tive Kalman filtering algorithms are used in many works ing is required for years or decades [7]. The holdover time [25]–[29] owing to a strong advantage: the current clock differs for different clocks being specified via the allowed state is predicted via its past state. A disadvantage is that fractional frequency departure that, for instance, in digital the Kalman filter needs the covariance of the clock noise, communication network nodes should not be greater than  which is commonly unknown to users. Moreover, Kalman claims the model to be known and all noises white Gauss1 × 10−11 [8]. In the short term (days and months), the holdover ian. Otherwise, under the model temporary uncertainty problem may be solved by predicting current errors via and non-Gaussian noises, the Kalman filter may produce the nearest past history. On a long-term basis (months biased and noisy estimates [30]. The errors can be reduced and years), internal resources may be exploited if addi- using finite impulse response (FIR) filters having inhertional resonances of the oscillator resonant system are ob- ent bounded input/bounded output (BIBO) stability and servable. In applications to the oven controlled crystal os- better robustness against temporary model uncertainties cillators (OCXOs), the long-term holdover approach was and round-off errors. Using FIR filters, measurement on developed in [9] as the modulational method. The method a horizon of N past points is processed to estimate the employs a high aging correlation of the fundamental and state at n. Extensive investigations of optimal FIR filteranharmonic frequencies of crystal resonators [10] to form ing of state space models have been provided by Kwon a synchronizing signal. Its application to an OCXO with et al. [31]–[33] resulting in the theory of receding horizon control [34]. Of importance is that gains of the unbiased Manuscript received August 31, 2007; accepted May 9, 2008. The work FIR filters do not depend on noise properties [20] and past was supported by the CONACyT Project, SEP-2004-C01-47732. Y. S. Shmaliy and L. Arceo-Miquel are with the Guanajuato Uni- system states [32],[35] that makes such filters simple for versity, Department of Electronics, Salamanca, Gto., Mexico (e-mail: applications. [email protected]). In this paper, we address an unbiased predictive FIR Y. Shmaliy is also with the Kharkiv National University of Radio filter of a class of discrete time-state space models asElectronics, Ukraine. Digital Object Identifier 10.1109/TUFFC.913 sociated with holdover problems in remote wireless clock Abstract—This paper addresses an unbiased p-step predictive finite impulse response (FIR) filter of the local clock Kdegree time interval error (TIE) polynomial model with applications to the global positioning system (GPS)-based clock synchronization. Generic coefficients are derived for a 2-parameter family of the polynomial filter gains. A generalization is provided for the p-step linear (ramp) gain allowing for close to optimal predictive filtering of the TIE. Basic holdover algorithms are discussed along with their most critical properties. Efficiency of the proposed filter in holdover is demonstrated by simulation and in real applications to GPS-based (sawtooth and sawtoothless) measurements of the TIE of a crystal clock.

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synchronization, such as those based on a global positioning system (GPS). The rest of the paper is organized as follows. In Section II, we discuss the clock time model and formulate the problem. In Section III, we derive the filter. Generic coefficients for a 2-parameter family of the polynomial filter gains are also given here. Section IV is devoted to linear prediction. Here a general p-step linear (ramp) gain function and noise power gain are derived. Holdover algorithms are discussed in Section V. Simulation is provided in Section VI. In Section VII, we apply the holdover algorithm to real GPS-based measurements of the TIE of a crystal clock. Finally, concluding remarks and generalizations are drawn in Section VIII.

II. Clock Time Model and Problem Formulation The discrete time-invariant model of a local clock can be represented in state space at a current time point n with the state and observation equations, respectively, as follows

l n = Al n -1 + w n ,

(1)



x n = Cl n + v n ,

(2)

where the K × 1 vector of the clock states xkn, k∈[1, K], is

T

l n = éë x 1n x 2n  x Kn ùû ,

(3)

in which x1n is the time interval error (TIE), x2n is the fractional frequency offset, x3n is the fractional frequency drift rate, etc. The K × K transition matrix



é1 ê ê ê0 ê A = êê 0 ê ê ê ê êë 0

t

t2 2

1

t 

0

1 

 0

 0 



t K -1 ù (K -1)! ú ú t K -2 ú (K -2)! ú ú t K -3 ú (K - 3)! ú

1



(4)

ú ú ú úû

C = éë 1 0 ¼ 0 ùû .

(5)

The K × 1 clock noise vector

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2008

Fig. 1. The p-step toward predictive FIR filtering problem for holdover.

contains zero-mean components with known covariance. In GPS-based synchronization, via 1PPS signals, the TIE excursions are compensated on a horizon of N points in the presence of the zero-mean measurement noise vn that commonly dominates. Let us now suppose that the 1PPS-based measurement ξn (2) has been provided from some far past point up to n – 1. At n, measurement is not available or wrong (unapplicable). The problem formulates as follows. We would like to derive an unbiased p-step predictive FIR filter of the clock first state x1 at n by processing the data taken from the time interval [n − N + 1 − p, n − p], p > 0, of the past history (Fig. 1). In such a formulation, the problem seems to be most general for holdover. We also would like to test the filter by simulation and GPS-based sawtooth and sawtoothless measurements of the TIE of a crystal clock. III. An Unbiased p-Step Predictive FIR Filter Below, following the diagram shown in Fig. 1, we derive a real time unbiased FIR estimator intended for p-step predictive filtering of the clock state x1n. A. General Relations To find the estimate, we start at n − N + 1 − p and represent (1) and (2), ignoring wn, as

projects a nearest past clock state λn−1 to the current state λn, where τ is a sampling time identical to 1  s in GPS-based clock synchronization using the one pulse per second (1PPS) signal. For the measurement ξn of the clock first state x1n used in synchronization, the 1 × K measurement matrix is

vol. 55, no. 10,

T

w n = éë w 1n w 2n ¼ w Kn ùû

(6)



l n = A N -1+ pl n -N +1- p,

(7)



x n = Cl n + v n ,

(8)

where



é 1 ti ê ê ê0 1 ê i A = êê 0 0 ê ê  ê ê êë 0 0

t 2i 2 2



ti



1 0

 

(ti) K -1 ù (K -1)! ú ú (ti) K -2 ú (K -2)! ú ú (ti) K -3 ú. (K - 3)! ú

1

(9)

ú ú ú úû

Now, the noiseless state model (7) projects ahead from n − N + 1 − p to n with the degree Taylor polynomial such that the first state (TIE) is represented with

Shmaliy and Arceo-Miquel :

K -1

x 1n =

å

x 1+q

q =0



efficient predictive estimator for holdover in gps-based clock synchronization

t q(N - 1 + p) q q!

= x 1 + x 2t(N - 1 + p) + x 3 ++xK

t 2(N - 1 + p) 2 (10) 2

(t(N - 1 + p)) K -1 , (K - 1)!

N -1+ p



å

h lix n -i

i =p W lT X,

=



(11)



W l = [h lp h l(p +1)h l(p +N -1)]T ,

(12)



X = [ x n - p x n - p -1x n - p -N +1]T .

(13)

N -1+ p

h li(N , p) = 1,

(14)

i =p

N -1+ p



å



v = éë v 1(n - p) v 1(n - p -1)  v 1(n - p -N +1) ùû . (18)

h li(N , p)i u = 0,

u Î [1, l ],

(15)

i =p

ensure that the prediction error is zero in the absence of noise when K ≤ l + 1.



To specify Wl requires examining the mean square error (MSE) of the predicted estimate (11),

J = E {(x 1n - x 1n ) 2}

(16a)



= E {(x 1n - W lT X) 2}

(16b)



W lT x

(16c)

where

J = (x 1n - W lT x) 2 + W lT RW l ,

-

W lT v) 2},

(19)

where the N × N covariance matrix R has a generic component Ri,j = E{vivj}, i,j ∈ [n − p, n − p − N + 1]. As can be seen, the first term in (19) represents the square bias and the second one the variance, both in the p-step toward prediction of the clock 1-state. It also follows that, for the prediction to be unbiased at any point, the following equality must be satisfied, namely, x 1n = W lT x.

(20)

To solve (20) for Wl, the well-known condition from the Kalman-Bucy filter theory may be applied, namely, the order of the optimal filter is the same as that of the system. That means that for x1n represented with (10) and the (K − 1)-degree polynomial, of the same degree l = K − 1 a polynomial1 must be used to describe the gain

h li =

åa jli j ,

(21)

j =0

where ajl are still unknown coefficients. Now, following [35], substituting (10) with K = l + 1, (12) with (21), and (18) in (20) yields the generic coefficient a jl = (-1) j



M (j +1)1 , |D|

(22)

calculated via the determinant |D| and minor M(j+1)1 of the (l + 1) × (l + 1) quadratic symmetric matrix



B. Generic Coefficients for Polynomial Gains

= E {(x 1n -

T

l

It is known that unbiasedness in the FIR estimate is guaranteed if hli, existing from p to N − 1 + p, obeys several fundamental properties [20], [35]. Modified for (11), these properties, namely

å

x = éë x 1(n - p) x 1(n - p -1)  x 1(n - p -N +1) ùû , (17)



where l, l ≥ 0, is the degree of the FIR filter polynomial gain hli. The l-degree N × 1 gain matrix and N × 1 matrix of measurements are given with, respectively,



T



By the commutativity property W lT v = vTWl and zero-mean noise, E{vn} = 0, we have

where x k  lx k(n -N +1- p), k ∈ [1, K] is the clock k-state at n − N + 1 − p. To provide the p-step toward predictive estimate x 1n  x 1n |n - p;N of x1n via (7) and (8) using N points starting at n − N + 1 − p, the discrete convolution operator can be applied to (8), x 1n =

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éd 0 d 1 ê d2 êd D=ê 1   ê ê d l d l +1 ë

 dl ù ú  d l +1 ú ú,  ú  d 2l úû

(23)

in which the component dr, r∈ [0, 2l] is defined with the Bernoulli polynomials Bn(x) [36] as N -1+ p

d r(N , p) =

å i =p

ir =

1 [ B r +1(N + p) - B r +1(p) ]. r +1 (24)

1 Unbiasedness is also achieved with the redundant degree polynomial, provided l ≥ K − 1, although with larger noise.

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Determined hli, the unbiased p-step predictive FIR filtering of the clock 1-state is provided at n (Fig.1) by (11). C. Noise Power Gain The estimate variance s n2 is evaluated by the term W lT RW l in (20). With large N, the noise induced by the GPS timing receiver becomes delta-correlated [35]. If also 2 its variance s vn is supposed to be constant over the horizon, then R becomes diagonal with equal components 2 2 s vn , and we have s n2 = g l s vn , where g l(N , p) = WlT Wl



(25)

Fig. 2. Linear (ramp) gains h1i. Filtering corresponds to p = 0 and predictive filtering to p > 0.

is the noise power gain (NG) [20]. For (15) and (21), the NG is defined to be N -1+ p



gl =

å

h li2

l

N -1+ p



(26a)



h 1i(N , 0) =

2(2N - 1) - 6i , N (N + 1)

(30)



h 1i(N ,1) =

2(2N + 1) - 6i , N (N - 1)

(31)

i =p



=

åa jl å j =0



j

h lii

i =p

= a 0l ,

(26b)

meaning that the coefficient a0l in (21) is responsible for noise in the unbiased estimate. As it will be shown below, a0l becomes zero when N tends toward infinity. IV. General p-Step Linear (Ramp) Gain Now, by (21)–(24), the gain of any degree l ≥ 0 of the unbiased predictive FIR filter can easily be derived for p > 0. It is known, however, that linear predictors are optimal or close to optimal for the prediction of clock instabilities [13]. In this particular case, l = 1, the ramp gain is specified with2

h 1i(N , p) = a 01(N , p) + a 11(N , p)i

(27)

with the coefficients

a 01 =



2(2N - 1)(N - 1) + 12p(N - 1 + p) , (28) N (N 2 - 1) a 11 = -

6(N - 1 + 2p) . N (N 2 - 1)

(29)

For small p∈[0, 3], (27) yields 2 For

[37].

the fixed-point problem, a function similar to (27) was found in



h 1i(N , 2) = 2



h 1i(N , 3) = 2

2N

2

+ 9N + 13 - 3(N + 3)i N (N 2 - 1)

2N

2

,

+ 15N + 37 - 3(N + 5)i N (N 2 - 1)

(32)

. (33)

We notice that the gain (30) earlier derived in [38] using linear regression and thereafter in [35] via the state space model is not predictive. The gain (31) was originally derived and investigated in [20] using the Lagrange multipliers. To figure out what happens when p increases, (30)–(33) are sketched in Fig. 2. An analysis reveals that (27) responds to p by increase on 6p/N(N + 1) in both peak values. For (26b) and (28), the NG is defined to be

g 1(N , p) =

2(2N - 1)(N - 1) + 12p(N - 1 + p) N (N 2 - 1)

(34)

and its particular values are provided by (30)–(33) if set to i = 0. Fig. 3 illustrates (34), manifesting that prediction is achieved at increase of noise. Indeed, when 2 ≤ N ≤ Nb, where Nb is determined by solving g1(Nb, p) = 1, the filter becomes inefficient, because noise is gained when NG exceeds unity. Clock synchronization, however, is commonly provided with large horizons. In this case, when N ≫ Nb > 2, the NG does not depend on p and fits the asymptotic function

g 1| N 1 @

4 N

(35)

Shmaliy and Arceo-Miquel :

efficient predictive estimator for holdover in gps-based clock synchronization

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Fig. 3. NG of the 1-degree (ramp gain) unbiased p-step predictive FIR filter for small p. The case of p = 0 is not predictive. An asymptotic line (dashed) corresponds to N ≫ 1.

dashed in Fig. 3. Moreover, for N  1 and (35), the estimate tends to be optimal in the sense of both zero bias and zero noise variance. V. Holdover Algorithms Based on the proposed filter, the holdover algorithm can be designed following one of the diagrams shown in Fig. 4. Here the measurement (input) is depicted with “○,” predicted estimate (output) with “●,” and unavailable or excluded “bad” measurement by “×.” It is implied that every measured point (from 1 to 9) represents the last point of the averaging horizon of N points shown in Fig. 1. From each of these points, the filter is able to produce the p-step toward predictive estimate. For example, the point 1 shown in Fig. 4(a) representing a horizon of N neighboring past points produces the estimate at 2a associated with the measurement at 2. The algorithm can be organized either with the fixed step, as shown in Fig. 4(a) and (b), or variable step, as shown in Fig. 4(c). In the fixed-step case of p = 1 shown in Fig. 4(a), the algorithm operates in “hold-in” (from 1 to 3) producing the estimates (from 2a to 4a). At 4, the measurement is not available. Therefore, a predicted value (4a) is used instead, and the filter produces the next estimate at 5a. Such a holdover procedure is applied from 4 to 6. The filter then returns to hold-in. In a like manner, the algorithm is organized for the case shown in Fig. 4(b), and one can easily figure out how to design the fixed-step algorithm for arbitrary p > 0. In the variable-step case illustrated in Fig. 4(c), the point 3 covers all the holdover range, producing individual estimates for the points 4a to 7a. Here, predicted values are not involved in holdover and are used in the subsequent hold-in range. VI. Simulation Before processing real measurements, we simulate a linear TIE function in the presence of discrete white Gauss-

Fig. 4. Holdover algorithms with predictive filtering: (a) fixed-step (p = 1), (b) fixed-step (p = 2), and (c) variable step.

ian noise as shown in Fig. 5 to realize the trade-off between the fixed-step, p = 1 [Fig. 4(a)], and variable-step, p = var [Fig. 4(c)], algorithms. Both algorithms were run with N = 500 covering the holdover range of 2000 points. Ten predictive estimates for consequently generated random entries are shown in Fig. 5(a) and (b) for p = 1 and p = var, respectively. Inferring that the prediction errors are similar and their more precise evaluation is a special topic, we investigate below only the fixed-step case. To ascertain limiting facilities of the fixed 1-step algorithm as shown in Fig. 4(a) in the presence of brightly pronounced model uncertainty, we generate and processed measurements of the linear TIE function x1n associated with the fractional frequency offset of x2 = 10−10. Simulation has been provided for the white Gaussian and uniformly distributed random measurement noises. To obtain holdover, with p = 1, the anomaly (rectangular jump) is excluded from the database. To provide filtering, with p = 0, the full database is processed. Even a quick look at Fig. 6 confirms the high efficiency of the solution. In fact, in the initial hold-in range, the values produced with p = 0 and p = 1 are indistinguishable. In the holdover range, the predicted estimate (p = 1) still does not get out of the actual linear behavior and keeps tracing it in the subsequent hold-in range. In turn, the filter (p = 0) exhibits inherent transients both within and beyond the holdover range. VII. Applications to GPS-Based Measurements The fixed-step, p = 1, holdover algorithm has been applied to GPS-based measurements of the TIE x1 of a local crystal clock imbedded in the Stanford Frequency Counter SR620 (Stanford Research System, Inc., Sunnyvale, CA).

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Fig. 6. Simulation of the unbiased (l = 1) 1-step (p = 1) predictive FIR filtering estimate x 1 of a linear TIE model x1 with different measurement noises, by N = 500: (a) white Gaussian and (b) uniformly distribFig. 5. The trade-off between the holdover predictive FIR filtering algorithms, by (30) and N = 500: (a) fixed-step, p = 1, and (b) variable step.

To provide measurement of x1, we use the SR620 and the 1PPS signal of the SynPaQ III GPS Timing Sensor (Synergy Systems, LLC, San Diego, CA). The Symmetricom cesium standard of frequency CsIII is used as reference to measure actual x1. To demonstrate holdover, an interval of measurements of about one hour is voluntarily excluded from the database. A. Sawtooth Measurements In the first experiment, we use measurements with the uniformly distributed sawtooth noise [39] induced by the SynPaQ III owing to the principle of the 1PPS formation. The fixed-step algorithm is applied with l = 1 and N = 2000 letting p = 1 or p = 500. Intentionally, we slightly shift the holdover range to obtain best and worst predictions. The results are sketched in Fig. 7(a) and (b), respectively. As can be seen, a real picture is similar to that obtained by simulation (Figs. 5 and 6). Larger errors occur, however, because of temporary GPS time uncer-

uted. The estimate xˆ 1 (p = 0) is provided with the 1-degree unbiased FIR filter. To demonstrate holdover, bad points are excluded from the data.

tainty. One may also observe that the estimate is highly insensitive to p when N is large, as it is stated by (35) and Fig. 3. In fact, the difference between the predicted estimates produced for p  = 1 and p = 500 results in Fig. 7(a) mostly in time shifting without substantial magnitude errors. More precisely, Fig. 8(a) demonstrates what happens if the predicted value is taken with a step p multiple to 100. In Fig. 8(b), we also show the relevant estimates found numerically via the p-step quadratic gain function. Larger noise and divergency in Fig. 8(b) confirm the near optimum quality of linear prediction. B. Measurements with Compensated Sawtooth In the second experiment provided in the same time scale, we apply the fixed-step holdover algorithm to measurements with corrected sawtooth. The results are shown in Fig. 9. Because the sawtooth correction works as a lowpass filter, the estimates obtained via the sawtoothless and sawtooth measurements appear to be closely related. That can be seen by comparing Fig. 7 and Fig. 9.

Shmaliy and Arceo-Miquel :

efficient predictive estimator for holdover in gps-based clock synchronization

Fig. 7. Unbiased predictive estimates x 1 of the crystal clock TIE x1, by N = 2000, p = 1, p = 500 and l = 1: (a) best case and (b) worst case. Measurements of the TIE were provided with sawtooth.

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Fig. 9. Unbiased predictive estimates x 1 of the crystal clock TIE x1, for N = 2000, p = 1, p = 500, and l = 1: (a) best case and (b) worst case. Measurements of the TIE are provided with sawtooth correction.

VIII. Conclusions In this paper, we proposed an unbiased predictive FIR filter applicable to holdover in real time remote wireless (GPS-based) clock synchronization. Unique coefficients were derived for a 2-parameter family of the l-degree polynomial filter gain functions. The case of linear predictive filtering, l = 1, has been investigated in detail as near optimal by simulation and via GPS-based measurements of the TIE of a crystal clock. Based upon the fixed-step and variable-step, holdover algorithms have been worked out and examined. Although the trade-off between these algorithms is certainly a special topic, their most important engineering features can now be sketched regarding the averaging horizon N ≫ 1 and holdover range Nh as follows:

Fig. 8. Unbiased predictive estimates x 1 of the crystal clock TIE x1, for N = 2000 and different p: (a) ramp gain, l = 1 and (b) quadratic gain, l = 2.

• With Nh < N, the variable-step algorithm produces smaller noise, although both algorithms yield similar results, on average (Fig. 5). • By Nh = N = p, the NG (34) associated with the ramp gain (27) is g1 ≅ 28/N , meaning that the variablestep algorithm becomes efficient when Nh = N > 28. • With N < Nh < N/2 ( N - 1/3 - 1 ), errors in both algorithms are closely related (Fig. 5). • With Nh > ( N - 3)/6, the variable-step algorithm becomes inefficient. • Both algorithms are stabilized by increasing N.

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Several other common inferences should also be mentioned: • Because the filter was derived in the zero-bias sense, its estimate represents the goodness-of-fit predictive test produced by regression. The latter, however, does not give us the filter gain function required for design and optimization of the clock synchronization system. • Unlike recursive IIR structures such as the Kalman filter, the FIR structure is inherently BIBO stable. This property appears to be of special importance when measurement noise is not white (sawtooth) and the model exhibits temporary uncertainty (GPS timeinserted). • With large N, the predictive estimates are insensitive to p; this property makes such filters useful in design of smart adaptive predictive FIR structures with variable horizons and prediction steps. We finally note that the filter proposed is simple in application. The algorithm does not require any knowledge about the noise, and our further investigations shall be focused on solving particular holdover problems and practical implementations.

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Systems were published by Springer. He was awarded the title Honorary Radio Engineer of the USSR in 1991; listed in Marquis Who’s Who in the World in 1998; and listed in Outstanding People of the 20th Century, Cambridge, England, in 1999. He is a member of several professional societies and organizing and program committees for international symposia. His current interests include the stochastic systems theory, precision resonators and oscillators, optimal estimation, and statistical signal processing.

Yuriy S. Shmaliy (M’96–SM’00) received the B.S., M.S., and Ph.D. degrees in 1974, 1976, and 1982, respectively, from the Kharkiv Aviation Institute, Ukraine, all in electrical engineering. In 1992, he received the Dr. Tech. Sc. degree from the Kharkiv Railroad Academy. In March 1985, he joined the Kharkiv Military University. He has served as full professor since 1986. In 1999, he joined the Kharkiv National University of Radio Electronics, and, since November 1999, he has been with the Guanajuato University of Mexico. Dr. Shmaliy has written 238 journal and conference papers and holds 80 patents. His books Continuous-Time Signals and Continuous-Time

Luis Arceo-Miquel was born in Mérida, Mexico, in 1973. He received the B.S. degree in digital systems engineering in 2001 from Merida Technological Institute of Mexico and the M.S. degree in electrical engineering in 2006 from the Guanajuato University, Salamanca, Mexico. Since 2007, he has been a postgraduate student of the Electronics Department of the Guanajuato University. His current research interests are in digital signal processing, GPS applications, instrumentation, and communications.