Fast and accurate estimation and adjustment of a local

Transcription

1 86 ieee transactions on ultrasonics ferroelectrics and frequency control vol. 53 no. 5 may 006 An Unbiased FIR Filter for TIE Model of a Local Clock in Applications to GPS-Based Timekeeping Yuriy S. Shmaliy Senior Member IEEE Abstract An unbiased finite impulse response (FIR) filter is proposed to estimate the time-interval error (TIE) K-degree polynomial model of a local clock in global positioning system (GPS)-based timekeeping in the presence of noise that is not obligatory Gaussian. Generic coefficients for the unbiased FIRs are derived. The low-degree FIRs and noise power gains are given. An estimation algorithm is proposed and examined for the TIE measurements of a crystal clock in the presence of the uniformly distributed sawtooth noise induced by the multichannel GPS timing receiver. Based upon this algorithm we show that the unbiased FIR estimates are consistent with the reference (rubidium) measurements and fit them better than the standard Kalman filter. I. Introduction Fast and accurate estimation and adjustment of a local clock performance making possible for a variety of modern digital systems to operate in common time with minimum slips is of importance for the global positioning system (GPS)-based timekeeping. Here the time interval error (TIE) between the GPS time and the local clock time is measured in the presence of noise induced by the GPS timing receiver. The TIE then is estimated and the correction signal adjusts the clock for the GPS time. The standard deviation of the noise using commercially available receivers is about 30 ns can reach 10 0 ns [1] and may be improved by removal of systematic errors to no less than 3 5 ns [1] []. Having such a large noise the measured data is usually not appropriate for clock correction and a TIE tracking filter with a large time constant is applied. To obtain filtering in an optimum way a TIE model of a local clock must be known for the filter memory. Such a model was proposed in [3] as the second degree Taylor polynomial and is now basic in timekeeping being practically proven. In the discrete time it may be written as: x 1 (n) =x 1 (0) + x (0)τn + x 3(0) τ n + w 1 (n τ) (1) Manuscript received April 6 005; accepted May This material was discussed December at the Precise Time and Time Interval (PTTI) Meeting Washington DC. The work was supported by the CONACyT Project J38818-A. Y. S. Shmaliy is with the Guanajuato University Salamanca Gto. Mexico ( shmaliy@salamanca.ugto.mx). He also is with the Kharkiv National University of Radio Electronics Kharkiv Ukraine. Fig. 1. Typical TIE 1 PPS sawtooth noise induced by a GPS timing sensor SynPaQ III. where n = ; τ = t n t n 1 is a time step multiple to the 1 s; t n is a discrete time; x 1 (0) is an initial time error; x (0) is an initial fractional frequency offset of a local clock from the reference frequency; x 3 (0) is an initial linear fractional frequency drift rate; and w 1 (n τ) is a random component caused by the oscillator noise and environment. In GPS-based measurements (1) is observed via the mixture: ξ 1 (n) =x 1 (n)+v 1 (n) () in which v 1 (n) is a noisy component induced at the receiver (noise of a measurement set is usually small). In modern receivers such as the Motorola M1+ (see [4]) and SynPaQ III GPS Sensor (Synergy Systems LLC San Diego CA) a random variable v 1 (n) is uniformly distributed owing to the sawtooth noise caused by a principle of the 1 PPS (one pulse per second) signal formation. Fig. 1 shows a typical structure of v 1 (n) in a short time; that is the modulo v max Brownian TIE associated with a phase of a receiver local oscillator where v max is the noise bound (in SynPaQ III it is about 50 ns. If the sawtooth correction is used the noise is reduced by the factor of about 5 and becomes near Gaussian). In long-term measurements v 1 (n) exhibits nonstationary excursions due to uncertainties in the GPS time [5] with different satellite sets in a view. To estimate optimally the states of different clocks (atomic and crystals) several filtering algorithms have been examined during a couple of decades. For the state space equivalent of (1) the problem was formulated by Allan and Barnes in [6] to apply Kalman filtering. The solutions then were given in [7] [8]. Thereafter the Kalman algorithm was used by many authors [9] [13] and its applications in time scales were recently outlined in [14]. The problem with the Kalman filter arises when noise is not white and the model demonstrates temporary uncertainties; thus the estimate may become biased and noisy. To overcome advanced Kalman algorithms were proposed in [15] [16] being however not yet adopted for GPS applications /$0.00 c 006 IEEE

2 shmaliy: proposed finite impulse response filter 863 An alternative approach is known as finite impulse response (FIR) filtering allowing for noise of arbitrary distribution. In contrast to the infinite impulse response (IIR) structures (including Kalman filters) FIR structures have inherent properties such as a bounded input/bounded output (BIBO) stability and robustness against temporary model uncertainties and round-off errors [17]. They may be used independently or combined with Kalman filters [1] [18]. A general optimal FIR filtering algorithm with embedded unbiasedness for state space models was recently proposed in [19]. Especially for GPS-based timekeeping and a linear TIE polynomial model an unbiased FIR filter was designed and studied in [0]. In this paper we propose a new unbiased FIR filter and algorithm intended for real time estimating the TIE K- degree polynomial model in the presence of a GPS noise of arbitrary distribution (with or without the sawtooth correction). The rest of the paper is organized as follows. In Section II we formulate the problem. In Section III a design of an unbiased FIR filter is given (the necessary coefficients are postponed in Appendix I). The low-degree FIRs and noise-power gains are derived in Section IV. An unbiased FIR-filtering algorithm for a single multichannel GPS timing receiver is described in Section V and applied for a local crystal clock without the sawtooth correction. The FIR estimates are compared here to the reference measurement (rubidium) and to those obtained using the 3-state Kalman filter (Appendix II). Concluding remarks are drawn in Section VI. II. Problem Formulation Most commonly the TIE polynomial model projects aheadonahorizonofn points from the start point n =0 with the K-degree Taylor polynomial: x 1 (n) = K p=0 x p+1 τ p n p p! + w 1 (n τ) = x 1 + x τn + x 3 τ n + x 4 6 τ 3 n 3 + w 1 (n τ) (3) where x l+1 x l+1 (0) l [0K] are initial states of the clock and w 1 (n τ) is a noise with known properties. By extending the time derivatives of the TIE model to the Taylor series (3) and () become respectively: λ(n) =A(n)λ(0) + w(n τ) (4) ξ(n) =Cλ(n)+v(n) (5) where λ(n) =[x 1 (n)x (n)...x K+1 (n)] T is a vector [(K + 1) 1] of the clock states and a time-varying clock matrix [(K +1) (K +1)]is: 1 τn τ n /... (τn) K /K! 0 1 τn... (τn) K 1 /(K 1)! A(n) = (τn) K /(K )! (6) The most common situation that may be assumed in timekeeping is when all or several clock states are observable by (5) via M (independent or dependent) measurements in the presence of correlated or uncorrelated noises. Then the observation vector is ξ(n) = [ξ 1 (n)ξ (n)...ξ M (n)] T and: c C = 0 c (7) c MM... 0 is a measurement matrix [M (K + 1)] with typically unit components c uu u [1M]. The clock noise vector is w(n τ) = [w 1 (n τ)w (n τ)...w K+1 (n τ)] T with the components caused by the oscillator noises. In view of Fig. 1 the noise vector v(n) =[v 1 (n)v (n)...v M (n)] T contains correlated or uncorrelated components that are not obligatory Gaussian. It is important that the GPS noise v(n) dominates on a horizon N; that is typically wu(n τ) N vl (n) N. Therefore w(n τ) maybeneglected in the averaging FIR procedure (it cannot be discarded in the Kalman filter). The problem now formulates as follows. Given a noisefree TIE model (4) w(n τ) = 0 that is observable via (5) in the presence of a mean zero noise v(n) =0distributed arbitrary with a known covariance v(n)v T (n). We want to derive a real-time unbiased FIR estimator of the clock states λ(n) supposing in this paper that M = K +1andC is an identity matrix. III. An Unbiased FIR Filter Using N points of the nearest past the FIR estimate ˆλ(n) =[ˆx 1 (n)ˆx (n)...ˆx K+1 (n)] T at n-th point is given by neglecting w(n τ): ˆλ(n) = N 1 H(i)ξ(n i) = q(n)γ = d(n)γ + r(n)γ (8a) where an auxiliary unit matrix (N 1) is Γ =[ ] T and the matrix [(K +1) (K + 1)] of unknown FIRs is: h K (i) H(i) = 0 h K 1 (i) (9) h 0 (i) in which the l-th FIR has inherent properties: h l (i) = { h l (i) 0 i N 1 0 otherwise and N 1 h l(i) = 1. The es-

3 864 ieee transactions on ultrasonics ferroelectrics and frequency control vol. 53 no. 5 may 006 timates vector and its deterministic and random constituents all of [(K +1) N] are respectively: q(n) =[H(0)ξ(n) H(1)ξ(n 1)...H(N 1)ξ(n N +1)] d(n) =[H(0)Cλ(n) H(1)Cλ(n 1)...H(N 1)Cλ(n N +1)] r(n) =[H(0)v(n) H(1)v(n 1)...H(N 1)v(n N +1)]. The mean square error (MSE) of ˆλ(n) is written as: J(n) = [λ(n) ˆλ(n)] T [λ(n) ˆλ(n)] =[λ(n) d(n)γ] T [λ(n) d(n)γ] + [r(n)γ] T [r(n)γ] producing the estimate bias and variance respectively: (10) (11) (1) (13) bias[ˆλ(n)] = λ(n) d(n)γ (14) var[ˆλ(n)] = [r(n)γ] T [r(n)γ]. (15) A. Generic Coefficients for the FIR of an Unbiased Filter The necessary and sufficient condition for the unbiased estimate follows straightforwardly from (14); that is: λ(n) =d(n)γ (16) providing the rule to derive the FIRs for the clock states: x 1 (n) WK T x (n)... = λ 1(n) WK 1 T λ (n)... (17) x K+1 (n) W0 T λ K+1 (n) where W l =[h l (0)h l (1)...h l (N 1)] T (18) x K+1 l (n) λ K+1 l (n) = x K+1 l (n 1)... x K+1 l (n N +1). (19) For the clock (K +1 l)-th state (17) thus yields a relation: x K+1 l (n) =W T l λ K+1 l(n). (0) It is seen that (17) is valid for arbitrary n. Setting n = 0 expressing the components in (19) with the l- degree polynomial by (3) and involving the property N 1 h l (i) = 1 we go from (17) to the unbiasedness (or deadbeat) constraint: FW l = G (1) in which G =[ ] T is of [(l +1) 1] and N 1 F = (N 1) () 0 1 l... (N 1) l We notice that (1) is inherent for any other unbiased estimators e.g. for the minimum variance unbiased (MVU) and best linear unbiased estimator (BLUE). It follows from Kalman filtering that an optimum estimate is achieved if the model degree is equal to a number of the filter states minus one. This means that the components in (18) also may be substituted by the l-degree polynomial such as: h l (i) = l a jl i j (3) j=0 where a jl are still unknown coefficients. Embedded (3) the constraint (1) becomes: DB = G (4) where the FIR coefficients matrix is B =[a 0l a 1l... a ll ] T and an auxiliary quadratic matrix [(l +1) (l +1)] is given by: d 0 d 1 d... d l d 1 d d 3... d l+1 D = d d 3 d 4... d l (5) d l d l+1 d l+... d l where the generic component d m = N 1 im m [0 l] is determined by the Bernoulli polynomials (Appendix I). An analytic solution of (4) yields the generic coefficients for (3): a jl =( 1) j M (j+1)1 (6) D in which D and M (j+1)1 are the determinant and minor of (5) respectively. Determined a jl and h l (n) the unbiased FIR estimate of the clock (K +1 l)-th state becomes 1 : ˆx K+1 l (n) = N 1 h l (i)ξ K+1 l (n i) (7a) = Wl T Ξ K+1 l (n) (7b) = G T (FF T ) 1 FΞ K+1 l (n) (7c) 1 The second reviewer mentioned that (3) can be written as W l = F T B; therefore (4) follows from (1) with D = FF T.Wenotice that (4) then may be solved by B =(FF T ) 1 G and (7b) rewritten as (7c).

4 shmaliy: proposed finite impulse response filter 865 where: ξ K+1 l (n) Ξ K+1 l (n) = ξ K+1 l (n 1).... (8) ξ K+1 l (n N +1) B. Estimate Noise The estimate noise variance (15) now may be rewritten as: var[ˆλ(n)] = [r(n)γ] T [r(n)γ] K = WK p T R p+1(n)w K p p=0 (9) where the autocorrelation matrix R l (n) of(n N) has a generic component R l (i j) = v l (i)v l (j) i j [n n N + 1]. Accordingly the estimate variance associated with (7b) calculates: σ K+1 l (n) =WT l R K+1 l(n)w l. (30) It is important that by large N the sawtooth noise becomes delta-correlated. This degenerates R l (n) tothe diagonal form with the components R l (i i) =σvl (i) and tends (30) toward: σ K+1 l (n) =σ v(k+1 l) (n)wt l W l (31) where σv(k+1 l) (n) is a variance of the noise perturbing the (K +1 l)-th clock state at nth point. IV. Applications to the Clock TIE Polynomial Model In applications K is identified for the filter memory on a horizon [0 N 1]) by the clock precision. Typically it is assumed that (3) fits atomic clocks by K 1 and crystal clocks by K. However K =3mayberequiredfor low-precision crystal clocks. Below we derive the unbiased FIRs for all these cases. A. Low-Order FIRs for Unbiased Filters Setting l =0 1 3 and using the coefficient d m (Appendix I) we first calculate (6). Then (3) leads to the unique FIRs namely: h 1 (i) = h 0 (i) = 1 N (3) (N 1) 6i (33) N(N +1) The sawtooth noise produced by SynPaQ III becomes practically delta-correlated if N = 1800 with τ =1sorN = 180 with τ =10s that corresponds to 0.5 hour of averaging. Fig.. Unique FIRs of the unbiased filter: h 0 (i) by (3); h 1 (i) by (33); h (i) by (34); and h 3 (i) by (35). An example is given for N = 60. h (i) = 3(3N 3N +) 18(N 1)i +30i N(N +1)(N +) (34) h 3 (i) = 8(N 3 3N +7N 3) 0(6N 6N +5)i N(N +1)(N +)(N +3) + 10(N 1)i 140i 3 N(N +1)(N +)(N +3). (35) Fig. sketches (3) (35) for N = 60. A uniform FIR (3) corresponds to simple averaging and is optimal in a sense of a minimum-produced noise. This FIR is practically proven to be reasonable in GPS-based common view measurements [5]. A linear FIR (33) was derived in [0] by using linear regression to compensate a bias of simple averaging. Its kernel starts with a maximum h (0) = (N 1) N(N+1) = 4 N > 0andgoestoaminimum N 1 h (N 1) = (N ) N(N+1) = N 1 N < 0 having zero at n 0 = N 1 h 3. Its special peculiarity is r = lim (0) N h = (N 1) that allows one to synthesize a FIR by saving r = for arbitrary N. It is surprising that the FIR synthesized in such a way is equal to that derived in [1] for the 1-step linear prediction on a horizon [1 N]. B. Noise-Power Gains The noise-power gain corresponding to the l-degree FIR is specified by (31) to be g l (N) =σ K+1 l /σ v(k+1 l) = W T l W l. Its values associated with (3) (35) are given below respectively: g 0 (N) = 1 N (36) (N 1) g 1 (N) = N(N +1) (37) g (N) = 3(3N 3N +) N(N +1)(N +) (38) g 3 (N) = 8(N 3 3N +7N 3) N(N +1)(N +)(N +3). (39)

5 866 ieee transactions on ultrasonics ferroelectrics and frequency control vol. 53 no. 5 may 006 Fig. 4. Structure of the (K+1)-state unbiased FIR filtering algorithm for the K-degree TIE polynomial model observable with a single GPS timing receiver. Fig. 3. Noise gains of the unbiased FIR filters: l = 0 by simple averaging (36); l = 1 by (37); k = by (38); and k = 3 by (39). Dashed lines are the upper bounds calculated by (40). Fig. 3 illustrates (36) (39) manifesting that unbiasedness is achieved at increase of noise. Indeed the curves for l>0 trace above the lower bound 1/ N associated with simple averaging (l = 0) that produces minimum noise (among all filters). It also follows that by large N the noise gain is performed by g l (N) = (l +1)/ N and thus traces below the upper bound: { (l +1)/ N N (l +1) gl (N) 1 N < (l +1). (40) V. An Unbiased FIR Filtering Algorithm for a Single Multichannel GPS-Timing Receiver We now consider an important practical case in which the measurement ξ 1 (n) ofatiex 1 (n) is obtained with a single multichannel GPS-timing receiver. Here observations of the higher-order states may be formed by increments of the lower-order estimates. An unbiased FIR filtering algorithm then is written as: ˆλ(n) =q(n)γ (41) ξ k (n) = 1 τ [ˆx k 1(n) ˆx k 1 (n 1)] k > 1 (4) where q(n) is given by (10) and the observation components for (10) k>1 are formed by (4). The algorithm is illustrated in Fig. 4. The clock first state estimate ˆx 1 (n) is obtained with h K (i) atahorizon of N K points. The observation ξ (n) for the second state x (n) then is formed using (4) by increments of ˆx 1 (n). Accordingly ˆx (n) is achieved with h K 1 (i)atahorizonof N K 1 points. Inherently the first accurate value of ˆx (n) appears at (N K +N K 1 )th point starting from n =0.The last state estimate ˆx K+1 (n) is calculated with h 0 (i) ata horizon of N 0 points using ξ K+1 (n) thatisformedinthe same manner as ξ (n). The first correct value of ˆx K+1 (n) appears at (N K + N K N 0 )th point. For the quadratic TIE model (K =crystalclocks) the 3-state unbiased FIR batch algorithm becomes by (7a) and (4): ˆx (n) = 1 τ ˆx 1 (n) = 1 1 j=0 1 h (i)ξ 1 (n i) (43) h 1 (j)[ˆx 1 (n j) ˆx 1 (n j 1)] (44) ˆx 3 (n) = [ˆx (n r) ˆx (n r 1)] τn 0 r=0 (45) where h (i) andh 1 (i) are given by (34) for N = N and (33) for N = N 1 respectively. For the linear TIE model (K = 1 atomic clocks) (41) and (4) simplify to the - state form of: ˆx 1 (n) = 1 1 h 1 (i)ξ 1 (n i) (46) ˆx (n) = [ˆx 1 (n j) ˆx 1 (n j 1)] τn 0 j=0 (47) where h 1 (i) is given by (33) with N = N 1. Each state also may be calculated using the matrix forms (7b) or (7c). Below as an example of application we use this algorithm to estimate the TIE model of an oven crystal clock embedded to the Stanford Frequency Counter SR60 (Stamford Research Systems Inc. Sunnyvale CA). The measurement is done with SynPaQ III and SR60 for τ = 1 s (GPS measurement). Simultaneously to get a reference trend the TIE of the same crystal clock is measured by SR65 (Stamford Research Systems Inc.) for the rubidium clock (Rb-measurement). Initial time and frequency shifts between two measurements then are eliminated statistically and a transition to τ = 10 s is provided by the data thinning in time. At the early stage the TIE model was identified to be quadratic K =andn l are

6 shmaliy: proposed finite impulse response filter 867 TABLE I Average Error (error) and Allan Deviation (σ) ofthe Estimate (Est) for 9.7 Hours and τ =10s: F is FIR and K is Kalman. Errors are Given for Rb-Measurements. x ns y 10 1 D /s Est error σ x(10) error σ y(10) error σ D (10) F K K-F determined for each estimate in the minimum MSE sense 3. We also compare the unbiased FIR estimates to those obtained with the 3-state Kalman filter (Appendix II). A. Several Hours Measurements In this experiment a short-term measurement of the TIE has been done during several hours [Fig. 5(a)]. The algorithm then was run. The horizons were identified for τ = 10 s to be N 1 = 155 or 0.43 hours N = 950 or.64 hours and N 3 = 860 or.39 hours for the Rbmeasurements. Thereafter we set the values of q s in the Kalman filter (Appendix II) to obtain the minimum MSEs for the FIR estimates. Fig. 5 and Table I illustrate these studies showing that the unbiased FIR estimates ˆx 1 (n) ˆx (n) and ˆx 3 (n) and the relevant Kalman estimates ˆx(n) ŷ(n) and ẑ(n) respectively are consistent with however some differences. It follows from Table I that the FIR filter works accurately. Fig. 5(a) shows that ˆx 1 (n) andˆx(n) trackthemean value of the GPS measurement and that their offsets from the Rb-measurement are coursed mostly by the GPS time uncertainty. In this experiment a maximum estimate error of about 60 ns was indicated between the 8th and 9th hours when a time shift in the 1 PPS signal has occurred. It follows [Fig. 5(b)] that ˆx (n) andŷ(n) fitwellthe weighted by 1/τ increments of the Rb-measurement. Even so there are two special ranges (dashed). In range I the frequency shift of about has occurred in the span between the 7th and 8th hours and no appreciable error is indicated in a range of large time shifts (between the 8th and 9th hours) in Fig. 5(a). We associate it with the frequency shift in SR65. In range II the Kalman filter demonstrates a brightly pronounced instability caused likelybythetemporarymodel uncertainty but the FIR estimate still is consistent. We watch for a bit shifted trends of ˆx 3 (n) andẑ(n) in Fig. 5(c) that may be explained by some inconsistency between the q s and N l. It also is seen that ẑ(n) traces much upper ˆx 3 (n) after about 8.7 hours. We associate it with the Kalman filter instability like the case of a range II in Fig. 5(a). 3 To identify K ˆx 1 (n) is compared to the reference (rubidium) measurement in the MSE sense by changing N for K [0 3]. The minimum MSE identifies K and determines N for the TIE estimate. Given K the other N l are determined in the same manner. (a) (b) (c) Fig. 5. Short-time measurement and estimation of the crystal clock TIE model with the 3-state unbiased FIR algorithm and the 3-state Kalman filter: (a) TIE (b) fractional frequency offset and (c) linear fractional frequency drift rate. The experiment was repeated for τ =1s.Theresults are presented in Table II to mention that on the whole the picture (Fig. 5) remains the same. The only principle point to notice is that the Allan deviations of all estimates are reduced by a large number of the points. The FIR and Kalman estimates behave here closer to each othereven though the former is still more accurate with its lower error and much lower Allan variance. B. Long-Term Measurements The same crystal clock was later examined during about.5 days using only the unbiased FIR filter. The measure-

7 868 ieee transactions on ultrasonics ferroelectrics and frequency control vol. 53 no. 5 may 006 TABLE II Average Error (error) and Allan Deviation (σ) ofthe Estimate (Est) for 9.7 Hours and τ =1s: F is FIR and K is Kalman. Errors are Given for Rb-Measurements. x ns y 10 1 D /s Est error σ x(1) error σ y(1) error σ D (1) F K K-F ments inherently show oscillations caused by day s variations in temperature [Fig. 6(a)] and like the previous case all FIR estimates fit well the Rb-measurement. Using ˆx (n) the temperature drift [Fig. 6(b)] was estimated to be about (14to4 C) and ˆx 3 (n) calculatesthe aging rate by ˆx 3 (n) = /day [Fig. 6(c)]. (a) VI. Conclusions In this paper we presented an unbiased FIR filter for the TIE K-degree polynomial model of a local clock. In contrast to the standard Kalman filter the proposed solution does not require the noises to be white and does not involve any knowledge about noises in the algorithm. Instead the FIR filter needs a length N l that is determined for a given clock using a reference source in the minimum MSE sense. The filter produces a noise with a variance that reduces as a reciprocal of N l. We notice that timekeeping operates with large horizons N 1 and thus one should not expect appreciable discrepancy between the optimum andunbiasedestimates. The trade-off between the 3-state unbiased FIR algorithm and the 3-state Kalman algorithm has shown their consistency. However as it was demonstrated experimentally the FIR filter produces a smaller error and a lower Allan variance for the sawtooth noise. Moreover it may be advanced further to be optimal in a sense of a minimum MSE that is currently under investigation. Appendix A Coefficients of Matrix (5) The coefficients for (5) are calculated by: d m = N 1 i m = 1 m +1 [B m+1(n) B m+1 ] where B n (x) is the Bernoulli polynomial and B n = B n (0) is the Bernoulli coefficient. For low orders B n (x) maybe found in the reference books. For high orders the following recurrent relation is valid: B n (x) =n B n 1 (x)dx + B n. (b) (c) Fig. 6. Long-term measurement and estimation of the crystal clock TIE with the 3-state unbiased FIR algorithm: (a) TIE (b) fractional frequency offset and (c) linear fractional frequency drift rate. Several low order coefficients d m are given below: d 0 = N N(N 1) d 1 = d = N(N 1)(N 1) 6 d 3 = N (N 1) 4 d 4 = N(N 1)(N 1)(3N 3N 1) 30 d 5 = N (N 1) (N N 1) 1 d 6 = N(N 1)(N 1)(3N 4 6N 3 +3N +1). 4 For large horizon N 1 the coefficients d m may be calculated by d m N 1 = N m+1 m+1.

8 shmaliy: proposed finite impulse response filter 869 Appendix B Three-State Kalman Filtering Algorithm In the state space the TIE model (1) is given by: x(n) y(n) = 1 ττ / x(n 1) 01 τ y(n 1) + w 1(n τ) w (n τ) z(n) 00 1 z(n 1) w 3 (n τ) x(n) =Ax(n 1) + w(n τ) and (5) becomes assuming a single receiver ξ(n) = [ 100 ] x(n) y(n) + v(n) z(n) ξ(n) =Cx(n)+v(n). The noises w(n τ) andv(n) are mean zero and jointly uncorrelated. The sawtooth noise v(n) has a uniform distribution p(v) =1/v max and correlated increments. Its white Gaussian approximation has a variance V = σv = 1 vmax v max v max v dv = vmax/3. The autocorrelation matrix of the white Gaussian noise w(n) is given by [19]: q 1 + qτ 3 + q3τ 4 0 Ψ = τ q τ + q3τ 3 8 q + q3τ 3 q 3τ 6 q τ + q3τ 3 8 q 3τ 6 q 3τ q 3τ q 3 in which the diffusion coefficients q s namely q 1 q and q 3 specify the white FM noise (WHFM) white random walk FM noise (WRFM) and white random run FM noise (RRFM) respectively in the τ-domain power law. The linear Kalman filtering algorithm reads as follows. Enter the q s R n 1 andˆx n 1 then calculate recursively: R n = AR n 1 A T + Ψ K n = R n C T (C R n C T + V ) 1 ˆx n = Aˆx n 1 + K n (ξ n CAˆx n 1 ) R n =(I K n C) R n. Acknowledgment The author would like to thank Dr. Raymond Filler of the U.S. Army Research Development and Engineering Command (RDECOM) CERDEC; Dr. Charles Greenhall of the Jet Propulsion Laboratory (JPL) California Institute of Technology; Dr. Judah Levine of the National Institute of Standards and Technology (NIST); and two anonymous reviewers for valuable comments and remarks. References [1] W. Lewandowski G. Petit and C. Thomas Precision and accuracy of GPS time transfer IEEE Trans. Instrum. Meas. vol. 4 no. pp Apr [] F. Meyer Common-view and melting-pot GPS time transfer with the UT+ in Proc. 3nd Annu. Precise Time and Time Interval Meeting 000 pp [3] D. W. Allan J. E. Gray and H. E. Machlan The National Bureau of Standards Atomic Time Scale: Generation Stability Accuracy and Accessibility. NBS Monograph 140 Time and Frequency: Theory and Fundamentals National Institute of Standards and Technology 1974 pp [4] R. M. Hambly and T. A. Clark Critical evaluation of the Motorola M1+ GPS timing receiver vs. the master clock at the United States Naval Observatory Washington DC in Proc. 34th Annu. Precise Time and Time Interval Meeting 00 pp [5] J. Levine Time transfer using multi-channel GPS receivers IEEE Trans. Ultrason. Ferroelect. Freq. Contr. vol. 46 no. pp Mar [6] D. W. Allan and J. A. Barnes Optimal time and frequency transfer using GPS signals in Proc. 36th Annu. Freq. Contr. Symp. 198 pp [7] P. V. Tryon and R. H. Jones Estimation of parameters in models for cesium beam atomic clocks J. Res. National Bureau of Standards pp [8] J. W. 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Neuvo FIR-median hybrid filters with predictive FIR structures IEEE Trans. Acoust. Speech Signal Processing vol. 36 no. 6 pp Jun Yuriy S. Shmaliy (M 96 SM 00) was born January He received the B.S. M.S. and Ph.D. degrees in and 198 respectively from Aviation Institute of Kharkiv Kharkiv Ukraine all in electrical engineering. In 199 he received the Doctor of Technical Sc. degree from the Railroad Academy of Kharkiv Kharkiv Ukraine. In March 1985 he joined the Kharkiv MilitaryUniversityKharkivUkraine.Heserved as professor beginning in In 1999 he joined the Kharkiv National University of

9 870 ieee transactions on ultrasonics ferroelectrics and frequency control vol. 53 no. 5 may 006 Radio Electronics Kharkiv Ukraine. Since November 1999 he has been with the Guanajuato University of Mexico Salamanca Gto. Mexico as a professor. Dr. Shmaliy has 178 papers and 80 patents. He was awarded a title Honorary Radio Engineer of the USSR in 1991; was listed in Marquis Who s Who in the World in 1998; and was listed in Outstanding People of the 0th Century Cambridge England in He is a member of several professional societies and organizing and program committees of International Symposia. His current interests include the statistical theory of precision resonators and oscillators optimal estimation and stochastic signal processing for frequency and time.