Recent Patents on Space Technology, 2011, 1, 134-149
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GPS-Based FIR Estimation and Steering of Clock Errors: A Review§ Yuriy S. Shmaliy* and Oscar G. Ibarra-Manzano University of Guanajuato, Electronics Department, Salamanca, Gto., Mexico Received: 31 March 2011; Revised : 24 June 2011; Accepted: 30 June 2011
Abstract: This paper refers to recent patents in GPS-based clock estimation and synchronization and observes advances in estimation and steering of clock errors with the finite impulse response (FIR) technique. The problem one meets here is caused by the GPS time temporary uncertainty and non-Gaussian sawtooth noise induces in the commercially available GPS timing receivers. It is complicated with the clock colored noise (flicker and random walks) often making the Kalman filter low efficient and requiring weighted averaging. We examine applications of optimal FIR filtering, prediction, and smoothing for polynomial and state space clock models. Optimal and unbiased FIR estimators are observed in line with the ladder, thinning, and Kalman-like algorithms. It is noticed that, for large averaging horizons featured to highly oversampled and slowly changing with time clock models, simple unbiased FIR solutions become virtually optimal. The trade-off with the Kalman filter is also discussed and we show that the unbiased FIR estimator ignoring noise and initial conditions is a better choice in terms of accuracy, stability, and robustness in solving many of the clock problems.
Keywords: GPS-based timekeeping, clock, time interval error, FIR estimator. 1. INTRODUCTION Modern digital information technologies require extremely high precision and accuracy for local timescales. To enable, the function of time dissemination is ordered to the Global Navigation Satellite Systems (GNSS) such as the Global Positioning System (GPS) (USA) [1], Global Navigation System (GLONASS) (Russia), Galileo (Europe), COMPASS (China), and IRNSS (India) measuring up to world standards for signals and systems. However, precision of the delivered time is limited with noise and temporary time uncertainty caused by different satellites in a view [2]. Provided the GPS reference time, errors in the local clocks of different levels of precision and accuracy are estimated and steered. If inaccuracy of the GPS time at the receiver fits the tolerance level, steering is provided utilizing simple techniques, such as simple averaging. Otherwise, precise correction and prediction of local clock errors cannot be achieved and optimal estimators are required. The problem with optimal estimation of the local clock state is coupled with the clock highly intensive colored noise (flicker and random walks). Among the available solutions, the optimal finite impulse response (FIR) estimator was shown in [3] to have the following important advantages. Contrary to the infinite impulse response (IIR) recursive structures such as the Kalman filter [4], the transversal FIR structures dealing with N measured points are inherently bounded input/bounded output (BIBO) stable [5] and have better robustness against temporary uncertainties and round-off errors. These structures have low sensitivity to noise. *Address correspondence to this author at the University of Guanajuato, Electronics Department, Salamanca, Gto., Mexico; Tel: +52 (464)647-9940, Ext. 2455; Fax: +52 (464)647-9940, Ext. 2311; E-mail:
[email protected] §
This manuscript is an extended and updated version of the paper: Y. S. Shmaliy, O. Ibarra-Manzano, Recent advances in GPS-based clock estimation and steering, Recent Patents on Space Technology, vol. 2, pp. 1040, 2010.
Moreover, they allow noise to be nonstationary and colored with any distribution and known covariance function [3]. A natural payment for these advantages, about N times larger consumption of the computation time, can commonly be ignored in clock synchronization, especially in Master Clocks. In this review, we refer to several most recent patents in GPS-based clock synchronization and error estimation and examine advances in solving these problems with optimal FIR estimators. The rest of the paper is organized as follows. In Section 2, we consider a typical GPS-based set to estimate and steer the local clock errors and refers to recent patents in practical implementation of a variety of estimation and steering solutions. The time interval errors of the clock are discussed in Section 3. Section 4 gives fundamentals of FIR filtering and discusses the problems solved employing this technique. Unbiased FIR filtering, prediction and smoothing are considered in detail in Sections 5 and 6. Section 7 is devoted to FIR filtering of clock errors in state space. Here, we also observe the ladder and thinning algorithms. The trade-off with the Kalman filter is outlined in Section 8 and applications are given in Section 9. Finally, concluding remarks are drawn in Section 10. 2. GPS-BASED MEASUREMENT SET A typical set for measurement, estimation, and steering of clock errors employing the one pulse per second (1PPS) signals of the commercially available GPS timing receivers [3] is shown in Fig. (1). In such receivers, the 1PPS output is generated by the local time clocks (LTCs) referred to the GPS time. To place the latter close to the absolute time, measurements are permanently provided in the United States Naval Observatory (USNO) for the coordinated universal time (UTC) of the USNO master clock (MC). In order to get an estimate of UTC (USNO MC) time derivable from a GPS signal, a set of UTC corrections is provided as part of the message signal. This broadcast message includes the time
GPS-Based FIR Estimation and Steering of Clock Errors
difference in whole seconds between GPS time and UTC (USNO MC). During 1996, this difference was 11 s [1]. The message also includes the rate and time difference estimate between GPS time and UTC (USNO MC) allowing for a receiver to calculate an accurate estimate of UTC (USNO MC) with the mission goal of 28 ns (1 sigma) [1]. Outside the selective availability (SA) induced for security purposes, the estimate may have an accuracy of about 10 ns in the root mean squares sense. Practically, the USNO has been successful in predicting UTC to within about 10 ns [1]. A real-time potential uncertainty of GPS time for UTC (USNO MC) stays therefore at a level of about 14-ns [1]. The standard deviation of GPS time available precision mostly depends on random errors in different onboard clocks and different satellites in a view. It can be achieved at a level of 3-5 ns [6, 7]. To obtain high accuracy, all time delays featured to channels and signal propagation are compensated in the set (Fig. 1) at the early stage. The main source of random errors that cannot be removed from the received signal below 3-5 ns [6, 7] is associated with the GPS time precision (b) limited by different satellites in a view at their nonstationary orbits. The receiver adds the sawtooth noise (c) bounded 10 3 and fLTC is the with ± , where [ns] = 2 fLTC [MHz] frequency of the LTC oscillator. In the receivers such as the SynPaQ III GPS Timing Sensor, this frequency is chosen to be 10 MHz and the bound is thus = 50 ns. It would be 14 ns if fLTC = 40 MHz, for example. Contrary to the error (d) in the GPS timing 1 PPS signal that is conventionally zero-mean with the uncertainty structure (b) and sawtooth noise (c), the time interval error (TIE) (e) of a local clock is inherently nonstationary and random, although having low intensity short-time noise. If a high resolution time interval counter is used such as the Stanford Frequency Counter SR620, then the difference between (e) and (d) represents the TIE measured in the local clock. The TIE has the form (f) or (g) if measurement is provided with and without sawtooth (using the negative sawtooth code supplied by the receiver), respectively. To smooth excursions and eliminate sawtooth in an optimum way, a digital estimator (Filter) of the clock TIE is used. If the filter is optimal or close to optimal in the minimum mean square (MSE) error sense, then its output (h) ranges most closely to the actual behavior (e) and the TIE of a locked clock becomes near stationary (i) with a small amount of random departures. 2.1. Current & Future Developments The measurement set (Fig. 1) implements the strategy of the local clock estimation and steering via the GPS 1PPS timing signals. So long as the GPS time is most available for users via the commercial receivers, many practical implementations rely namely on GPS time. Depending on accuracy and precision required, different kinds of clocks (electronic, electromechanical, crystal, and atomic) are used in modern digital and analog systems and a variety of simple and sophisticated synchronization units have been invented 1877-6116/11 $100.00+.00
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and implemented for decades employing estimators meeting the tolerance requirements. In a locked clock time scale, noise mostly depends on precision of a local oscillator and time error departures are limited by the accuracy of reference time signals. High synchronization accuracy can be achieved here utilizing different kinds of the clock error estimators (filters), the design of which can also vary. Averaging and low-order low-pass (LP) filters are typical for commercially manufactured GPSDOs. It has been proposed in [8] to use averaging and integrating filters along with the phase locked loop (PLL). In [9], the use of the 3-order PLL was examined in order to minimize phase errors. A linear least squares estimator is exploited in [10] to discipline a rubidium standard. Some authors propose employing the Kalman filter [11] and even neural networks [12]. In recent years, the optimal and unbiased FIR structures have been developed for clocks in [3,13-24] owing to their ability to outperform the Kalman filter and several strong engineering features. These and other solutions outline current developments residing in a number of practical devices patented by the authors [2557]. Experimental investigations of several GPSDOs utilizing different filters can be found in [58, 59]. Examining the current state of the GPS-based clock estimation and synchronization, one may arrive at a conclusion that further progress can be achieved here by utilizing optimal and unbiased FIR filters and estimators. Namely such structures most naturally suit the clock model having colored noise components and measured in the presence of non-Gaussian noise induced by the GPS timing receiver. FIR estimators can be applied to both the polynomial and state-space models allowing noise to have any distribution and known covariance function and demonstrating the above-listed important advantages: inherent BIBO stability, robustness, and low sensitivity to noise and initial conditions. To make it feasible, below we observe in detail the methods of FIR filtering and steering of clock errors and sketch the guide for users. 3. CLOCK TIME INTERVAL ERROR The variation in the time difference between a real clock and an ideal uniform time scale is known as the TIE [60]. If we start at zero ( t = 0 ), then the TIE can be modeled at the present continuous-time point t using the finite Taylor series expansion as [61] t
x(t) = x(0) + y(0)t +
z(0) 2 t + wPM (t) + wFM ( )d , 2 0
(1)
where x(0) is the initial time error at zero, y(0) is the fractional frequency offset at zero of the local clock oscillator, z(0) is the linear frequency drift rate at zero, wPM (t) is the clock oscillator phase noise, and wFM (t) is the clock oscillator frequency noise. The IEEE Standard [60] suggests that the three state clock model such as (1) serves on the infinite time interval and thus the higher order clock states should be included to the random part. Note that the clock high order states, commonly insignificant for practical use, may cause errors in the estimates. © 2011 Bentham Science Publishers
136 Recent Patents on Space Technology, 2011, Volume 1, No. 2
Shmaliy and Ibarra-Manzano
USNO Time
(a)
GPS
Error GPS Time
(b)
Uncertainty
+ (c)
GPS Timing
Sawtooth Noise
(b)
Receiver
TIE (c)
(d)
Measurement Noise TIE of the
(d)
Unlocked
TIE Counter
Local Clock
(e) TIE
(e)
Measurement (f)
with Sawtooth
(g) (f)
Filter
TIE Measurement with Sawtooth Correction
(h) Local Clock
(g)
Filtered TIE
(e)
(i)
(h) (i)
Fig. (1). A typical set for GPS-based measurement, estimation, and steering of clock errors, after [3].
TIE of the Locked Local Clock
GPS-Based FIR Estimation and Steering of Clock Errors
Recent Patents on Space Technology, 2011, Volume 1, No. 2
As it follows from Fig. (1), the TIE is measured by the time interval counter over a time interval starting at t and ending at t . Therefore, TIE measurement is commonly represented in discrete time n. The clock error (1) can also be represented with digital numbers xn to model TIE over a time interval of some discrete points, from zero to n , as
xn = x0 + y0 n +
n z0 2 2 n + wPMn + wFMi , 2 i=0
(2)
where x0 = x(0) , y0 = y(0) , and z0 = z(0) . In precise clocks,
y0 and z0 are negligibly small and the deterministic part in (2) is therefore slowly changing with time. On the other hand, both wPMn and wFMn contain fast white and slow colored (flicker and random walks) Gaussian components. But even if we suppose that wFMi is white, then the last term in (2) represents the discrete-time Wiener process, making (2) nonstationary and near Markovian in the presence of phase noise and the environmental influences (Fig. 2). Owing to this, estimation of clock state has the following specifics: •
•
Short base of 10 2 …10 3 s (“Filtering of TIE” in Fig. 2) is assigned for the synchronization of crystal and rubidium clocks. Here the slow noise components are commonly tracked and the fast ones are filtered out. 5
Long base of about 10 s (“Filtering of xn ” in Fig. 2) is required for filtering and prediction in atomic clocks. Maximum accuracy is achieved here if most of the noise components and the environmental effects are filtered out.
•
Measurement on a very long base of 10 7 s is commonly processed to estimate the clock noise structure in terms of the Allan variance.
•
All the data available are processed to estimate the current clock state. TIE
137
random walks) and high-order clock states removed to the random part often make applications of the Kalman filter problematic. Moreover, in GPS-based timekeeping via 1PPS timing signals, the measurement noise is non-Gaussian (uniformly distributed). But even if accuracy is not an issue and the Kalman filter is chosen as an estimator for clock, the problem still remains of how to describe correctly the clock noise covariance matrix which is often unknown for customers. That could be provided via the clock noise power law or the Allan variance using special algorithms [62, 3]. However, the relevant values are commonly available only for a certain type of clock oscillators and only at some key points. On the other hand, in order for the Kalman filter to be maximally accurate (it is suboptimal anyway, owing to the clock nonwhite noise components), the covariance function of a particular clock has to be measured and then represented in white Gaussian approximation with a maximum accuracy that imposes one more problem. Based on this brief analysis, one may truly deduce that the Kalman filter is not a completely “natural” estimator for clocks and some more appropriate engineering solutions should be found. Below, we shall show that FIR filtering associated with weighted averaging serves better in solving many of the clock problems. 4. FIR ESTIMATION OF CLOCK STATE What is the FIR filter and how can it be useful in estimating the clock TIE model in different sense? Let us start with the discrete-time measurement
sn = xn + vn
(3)
where xn represents the clock TIE and vn is the zero mean, E{vn } = 0 , additive noise. The FIR filtering estimate xˆn of
xn is the weighted average of sn over N 2 points via the discrete convolution of sn and the FIR filter impulse response hln hln (N ) , where l is the degree of the impulse response [14] to be formally defined below. In the batch form, the FIR estimate thus reads N 1
Filtering of xn
xˆn = hli sni
(4)
= WlT S N = STN Wl .
(5)
i=0
xn
xn + noise
where the measurement vector S N N and filter gain matrix Wl N are specified with, respectively,
Flicker plus
environment
S N = [sn sn1 …, snN +1 ]T ,
Filtering of
Wl = hl 0
TIE
(6) T
hl1 hl ( N 1) .
(7)
Note that there exists some estimate error n = xn xˆn ,
0 m n N 1
n
Time
Fig. (2). Typical filtering problems in clocks.
It has to be remarked now that (2) augmented with the clock highly intensive nonwhite Gaussian noise (flicker and
caused by noise vn that needs to be minimized in some sense, by optimizing hln or Wl . The minimum MSE sense is commonly used, albeit it is not always appropriate for all clock applications.
138 Recent Patents on Space Technology, 2011, Volume 1, No. 2
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4.1. Problems Solved with FIR Structures
4.1.3. Smoothing
The FIR filtering approach represented by the convolution (4) allows solving the following estimation problems.
Smoothing is useful for noise reduction in postprocessing, identification of clock models, and ascertaining the initial clock state. Most generally, p-lag FIR smoothing can be organized as shown in Fig. (3c) to have a fixed lag ( p is constant), fixed interval ( N is constant), or fixed point ( n p is constant). The FIR smoother and smoothing FIR filter can be designed based on the following convolution forms, respectively,
4.1.1. Filtering Filtering is commonly used in solving clock problems to estimate the present state. To make it possible based on FIR structures, a nearest past history including the present point n is processed from n N + 1 to n and the estimate is related to n , by (4) or (5), as shown in Fig. (3a).
N 1
x n+ p = hli ( p)sni , p < 0 ,
N points
…
n – N 1
x n =
n
…
n
…
p >0
(b)
5. UNBIASED FIR FILTERING OF TIE
N points +
p-lag FIR Smoothing
…
n
The unbiased FIR filtering estimate (p = 0) of TIE (unbiasedness is required by the IEEE Standard [60]), can be said to be optimal in the zero-bias sense. To find the relevant impulse response hln , one may proceed with the unbiasedness condition [3]
Time p 0 , (Fig. 3b) is provided, if to organize the convolution batch as N 1
p > 0,
(5)
i=0
where hli ( p) hli (N, p) = hl (n+ p) (N, p) . There can also be designed the predictive FIR filter to obtain the estimate at n via the past history from n N + 1 p to n p as
x n =
N 1+ p
hli ( p)sni ,
(8)
meaning that the mean value of the estimate is required to be equal to that of its origin. Substituting (3) to (4) and then averaging the both sides of (4) by (12) gives us N 1
4.1.2. Prediction
x n+ p = hli ( p)sni ,
(11)
Note that the estimates (4) and (8)-(11) have practical usefulness only if the impulse responses are specified and optimized in some sense on an interval of N points. The number N should also be optimized, Nopt [16]. Several ways of optimization are discussed below.
n p Time
p-step FIR prediction
…
hli ( p)sni , p < 0 ,
As can be seen, the difference between (8) and (10) as well as between (9) and (11) exists only in the sign of p . By p = 0 , all of the p-shift estimates, (8)-(11), become (4).
(a)
n– N+ 1
i= p
N points n – N 1
N 1+ p
Time
FIR filtering N points
(7)
i=0
p > 0,
i= p
where hli ( p) hli (N, p) .
(6)
xn = hli xni
(9)
i=0
that, specifically for the deterministic part of the clock model (2), results in
z0 2 2 n 2 N 1 N 1 N 1 z = x0 hli + y0 hli (n i) + 0 2 hli (n i)2 . 2 i=0 i=0 i=0 x0 + y0 n +
(10)
By equating the terms in (14) and then accounting the identity associated with x0 in the remaining ones, one arrives at the fundamental properties of the unbiased FIR filters [14]: N 1
h
li
= 1,
(11)
i=0
N 1
h i li
i=0
u
= 0 , 1 u l,
(12)
GPS-Based FIR Estimation and Steering of Clock Errors
Recent Patents on Space Technology, 2011, Volume 1, No. 2
that can be rewritten in a short matrix form as
WlT V = JT ,
(13)
where J = [10…0 ]T , WlT is given by (7), and V is the Vandermonde matrix given below for arbitrary l as
V=
1 0 0 1 1 1 1 2 22 1 N 1 (N 1)2
… 0 … 1 … 2l … (N 1)l
.
(14)
To solve (17) for unknown Wl , one can observe that the T
inverse (V V)
1
exists. Then multiplying the right-hand side
of (17) with the identity matrix (VT V)1 VT V and then removing nonsingular V from both sides give us the unbiased FIR filter gain matrix [14, 63]
WlT = JT (VT V)1 VT .
(15)
It can be shown that (19) represents the first row of the familiar minimum variance unbiased (MVU) estimator [64], in which the observation matrix is Vandermonde (18).
139
Using (21)-(23), the impulse response hln (20) can be found analytically for any degree l . 5.1.1. Low-Degree Polynomial Impulse Responses For low-state models, the unique polynomial impulse responses of the unbiased FIR filters were found in [14] to be
h0n =
1 , N
(20)
h1n =
2(2N 1) 6n , N(N + 1)
(21)
h2n =
3(3N 2 3N + 2) 18(2N 1)n + 30n 2 . N(N + 1)(N + 2)
(22)
The response (24) represents simple averaging widely used in atomic clocks for noise reduction. The ramp response (25) was originally derived in [14] via linear regression, once the latter is used to model systematic behaviors in clocks [60]. Fig. (4) sketches several low-degree impulse responses, which applications in GPS-based timekeeping can be found in [3,13-23]. hln
5.1. Polynomial FIR Filter Gain
h3n
Although (19) is an exact and simple solution, it does not give us an idea about the gain function. However, one knows from the Kalman filter theory [4, 65] that the order of the optimal filter is the same as that of the system. Hence, the mdegree TIE polynomial function xn can unbiasedly be filtered if the polynomial impulse response hln has the same degree l = m. Note that unbiasedness is also achieved if l > m , although with larger noise. Referring to this fact, hln can be represented with the following polynomial [13], l
hln = a jl n , j
(16)
in which the generic coefficient a jl defined as
M ( j+1)1
via the determinant | D | and minor M ( j+1)1 of a short
(l + 1) (l + 1) quadratic symmetric matrix
d0
d1
…
d1
d2
… dl+1
dl
dl+1 … d2l
dl
,
0
N–1
Fig. (4). Unique low-degree impulse responses of the unbiased FIR filters.
dr (N ) = i r =
1 [ Br+1 (N ) Br+1 ] . r +1
where v2 is the variance of the white measurement noise vn and x2 is the variance of the estimate. NPG can also be evaluated via hln as [66] N 1
(23)
i=0
(18)
which component dr , r [0, 2l] , is calculated using the
i=0
n
1/N
gl = hli2 = WlT Wl = a0l
Bernoulli polynomials Bn (x) as N 1
h0n
The noise amount in the unbiased FIR filter output can be estimated with the noise power gain (NPG) gl = x2 / v2 , (17)
|D|
T D=V V=
h1n
5.1.2. Noise Power Gain
j=0
a jl = (1) j
h2n
(19)
stating that in order to reduce noise in the estimate, one should use a solution with a minimum a0l in (20) that is achieved by increasing N. It follows from (20) and (21) that NPG can also be found as gl = hl 0 . Fig. (5) sketches gl for several low-degree unbiased FIR filters. A shadowed area is limited with simple averaging. This is a dead zone that theoretically can never be crossed and its bound is given by 1 / N that is associated with (24).
140 Recent Patents on Space Technology, 2011, Volume 1, No. 2
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Note that simple averaging is optimum in the sense of the minimum produced noise among all other linear filters [67] applied to white Gaussian noise. 10
the variance of the measurement noise. If yn and zn are not available at n , one can substitute them with the maximum values featured to particular clocks. Then (29) and (30) would serve as lower bounds for N opt .
0
2/ N 3/ N 4/ N
5.1.4. Optimized Ramp Gain
l=3 10
Any unbiased impulse response with l = m can be used in order to obtain the estimate with a minimum bias. However, that does not mean that MSE is minimum, because the estimate variance is still not minimized. An optimization of the ramp response can be achieved in the minimum MSE sense if we adjust the coefficients a jl in (20) in order to
1/ N
-1
where a means a maximum integer smaller than a, y and z are the second and third clock states, respectively, and v2 is
l=2 l=1
g (N ) l
reduce the estimate noise variance at expense of bias. The gain optimized in such a way was found in [18] to be
h1n ( ) = Fig. (5). NPGs of the low-degree unbiased FIR filters.
It has been shown in [14] that gl (N ) traces above the lower bound 1 / N and below the upper bound specified with, provided l > 0 ,
l +1 , N (l + 1)2 1 , l > 0. < gl (N ) < N N 2 1 N < (l + 1)
(24)
It follows straightforwardly from (28) that noise in the estimate can be attenuated substantially if N is allowed to be large.
1 N
3(N 1) 6n , 1 + N +1
where is the optimization coefficient. As can be seen, (31) becomes unbiased (25), if to set = 1 , and = 0 simplifies it to simple averaging (24). The optimal value of 0 ,
0 < 0 < 1 , was specified in [18] in the minimum MSE sense as
0 =
2 N(N 2 1) 2 N(N 2 1) + 12
rubidium, and cesium clocks, 0 practically ranges as, respectively,
0.992124198176391 < 0 < 0.999999999964719,
Although the IEEE Standard [60] suggests that the clock has only three states, a general m-degree polynomial can also be used to model clocks. An analysis of Fig. (2) shows that the l -degree impulse responce fits the clock model only on an interval of N points, when l m . If l < m , bias occurs in the estimate and one needs to limit N with some maximum value allowing for the minimum MSE. Because N depends on the sampling time , a minimization of MSE by is also required.
0.998451946589028 < 0 < 0.999999993798186,
2 2 N opt (l = 0) = 3 2 v2 , y 144 2 N opt (l = 1) = 5 4 2 v z
,
(25)
(26)
(28)
with 2 = 2 y 2 / v2 . It was also shown that, for crystal,
5.1.3. Optimum Sampling and Averaging Intervals
It has been shown in [16] that the optimal sampling time opt is 1 s for GPS-based measurement of TIE via 1PPS timing signals employing the commercially available receivers. In turn, N opt for the uniform response (24) and ramp response (25) can be determined by, respectively,
(27)
0.969923570257806 < 0 < 0.999950387946414, meaning that, for different clocks, 0 is almost unity and the difference between the unbiased and optimal estimates is practically indistinguishable. Recall that the optimal ramp filter (31) requires four coefficients, , y , v2 and N , whereas the unbiased one (25) needs only N . One thus may arrive at the conclusion that there would be no reasonable necessity in using optimal filters to solve many of the clock problems. 6. UNBIASED FIR PREDICTION AND SMOOTHING It follows from (8)-(11) that prediction and smoothing are provided with the same convolution batch if to let p be either positive or negative, respectively. To organize unbiased predictive FIR filtering (9), by p > 0 , and smoothing FIR filtering (11), by p < 0 , the Vandermonde matrix needs to be modified as shown in [3],
GPS-Based FIR Estimation and Steering of Clock Errors
1 p p +1 1 V ( p ) = 1 p+2 1 N 1 + p
, … ( N 1 + p )l
… … …
pl ( p + 1)l ( p + 2)l
Recent Patents on Space Technology, 2011, Volume 1, No. 2
Note that, if p < 0 , this impulse response can also be employed in the p -lag ramp smoother. (29)
to obey the following fundamental properties of the unbiased impulse response, existing from p to p + N 1 : N 1+ p
hli ( p ) = 1
(30)
hli ( p)i u = 0 , 1 u l .
(31)
i= p
N 1+ p
141
i= p
with V( p) specified by (33), the gain (19) can now serve to provide prediction and smoothing.
A payment for prediction is instability. In fact, both (38) and (39) tend toward infinite if N 1 that makes prediction inefficient when gl ( p) exceeds unity (noise is not attenuated) with small N . The latter is neatly seen in Fig. (6). On the other hand, with N >> 1, the filter is highly efficient and its NPG fits the asymptotic function gl 4 / N (dashed in Fig. 6) that does not depend on p . Fig. (7) gives us an idea about how well the ramp FIR predictor works, covering the range of about 0.5 hours via the GPS-based measurements of the TIE [3]. As can be seen, x n slightly gets away from the actual behavior xn in the holdover range, when measurement is temporary unavailable, and then converges to xn in the hold-in range. Different prediction algorithms with examples of applications for such measurements can be found in [3].
6.1. Polynomial Solution If we now substitute (33) to (22), then the p -shift components in D( p) can be specified via the Bernoulli polynomials with [20] dr (N , p) =
N 1+ p
i= p
B (N + p) Br+1 ( p) i = r+1 . r +1 r
6(N 1 + 2 p) N(N 2 1)
6 1
0
35
10 3
p=0 10-1
10-2
100
102
1
10
103
104
N
Fig. (6). NPGs of the p-step predictive unbiased ramp FIR filter.
150
xn + noise
N = 500
(33)
with the p -shift coefficients found in [20] to be
a11 ( p) =
2 10
The impulse response of the p -step ramp predictive FIR filter has a familiar form of
2(2N 1)(N 1) + 12 p(N 1 + p) , N(N 2 1)
101
(32)
6.1.1. The p-Step Predictive Ramp FIR Filter
a01 ( p) =
g (N ) l
p = 60
20
Provided D( p) , the p -shift response hln ( p) can also be found analytically for any l and p . Smoothing, however, is not widely used in clock problems. Just on the contrary, error prediction occupies an important place. Moreover, linear predictors are proved to be optimal or close to optimal for the prediction of clock instabilities [35]. Therefore, below we consider only the p -step predictive ramp FIR filter, noticing that a quadratic solution can be found in [3, 20].
h1n ( p) = a01 ( p) + a11 ( p)n
102
100
~ x
xn n
(34) 50
(35) 0
and NPG specified with g1 ( p) = a01 ( p) as (38). By p = 0 , (37) becomes (25). For p = 1 , this response was originally derived and investigated in [66] using the Lagrange multipliers. An analysis reveals that an increase in p results in a higher slope of h1n ( p) as well as in an increase on 6 p / N(N + 1) in both positive and negative peak values.
0.2
0.4
0.6
0.8
Time, hours – 50
Fig. (7). Typical linear prediction of clock errors with unbiased FIR filters.
142 Recent Patents on Space Technology, 2011, Volume 1, No. 2
Shmaliy and Ibarra-Manzano
7. FIR FILTERING IN STATE SPACE
X n,m = A nm x m + Bnm N n,m ,
(42)
In line with the polynomial form (2), state space modeling is also widely used in solving clock problems.
S n,m = C nm x m + G nm N n,m + Vn,m ,
(43)
where 7.1. Clock Characterization in State Space
T
Allan and Barnes were the first suggested representing the clock in state space [2]. Fitting (2), the clock timeinvariant state and observation equations become, respectively,
X n,m = xTn xTn1 …xTm ,
(44)
S n,m = [ sn sn1 …sm
(45)
]
T
, T
x n = Ax n1 + w n ,
(36)
N n,m = wTn wTn1 …wTm ,
(46)
sn = Cx n + vn ,
(37)
Vn,m = [vn vn1 …vm ]T ,
(47) T
where the state vector x n 3 is given by
A i = (A i )T (A i1 )T …AT I ,
T
x n = [xn yn zn ] ,
(38)
the quadratic clock matrix A
1 /2 A= 0 1 0 0 1 2
33
,
(39)
projects the nearest past state x n1 to the present state x n via the finite Taylor series expansion, and the observation matrix C 13 is C = [10 0 ] . Vector w n 3 of the clock zero mean Gaussian noise,
w n = [wxn wyn wzn ]T ,
(40)
has the covariance matrix = E{w i wTj } . Supposing the clock noise components to be white Gaussian, this matrix can be approximated via the phase noise power law of the clock oscillator [60] as suggested in [62, 3],
q 2 q 4 q1 + 2 + 3 3 20 q2 q3 3 = + 2 8 q3 2 6
q2 q3 3 + 2 8 q3 2 q2 + 3 q3 2
q3 2 6 q3 2 q3
,
1 i Ai = 0 1 0 0
Bi =
A0 0 0 0
Ci =
(A i )1 (A i1 )1 , 1 (A )1 (A 0 )1
(A i )1 = 1 (41)
where the diffusion coefficients q1 , q2 , and q3 represent the white FM noise (WHFM), white random walk FM noise (WRFM), and white random run FM noise (RRFM), respectively. Noise vn associated with GPS-based measurement and composed by the GPS time temporary uncertainty and sawtooth induced by the receiver is commonly zero-mean, E{vn } = 0 , with the variance
E{vn2 } = v2 . An analysis of this noise can be found in [3]. 7.1.1. Representation on an Averaging Interval of N Points To apply FIR filtering to the clock state-space model, (40) and (41), the latter needs to be represented on an interval of N past point from m = n N + 1 to n (Fig. 2) as shown in [3]
( i)2 2 i 1
A A0 0 0
,
(49)
… A i1 … A i2 … A0 … 0
i
(I) (A) 1 1 0 (I)1 Gi = 0 0 0 0
(48)
Ai A i1 A A0
,
(50)
(51)
( i)2 / 2 , … (A i1 )1 … (A i2 )1 …
(I)1
…
0
(52)
(A i )1 (A i1 )1 , (A)1 (I)1
(53)
where (Z)1 means the first row of Z and 0 is a matrix with all components equal to zero. Note that the initial state x nN +1 in (48) is supposed to be given exactly. Therefore,
w nN +1 in (50) is always zero valued. Based upon (46), the clock states can now be represented at n via the states at n N + 1 as follows:
xn
= xnN +1 + (N 1) ynN +1 +
, (N 1)2 2 znN +1 + wxn 2
(54)
GPS-Based FIR Estimation and Steering of Clock Errors
Recent Patents on Space Technology, 2011, Volume 1, No. 2
143
yn = yn N +1 + ( N 1) zn N +1 + wyn ,
(55)
WlT0 C nm = (A N 1 )1 ,
zn = zn N +1 + wzn ,
(56)
where wxn , wyn , and
wzn represent the noise components on
where the gain Wl 0 is exactly that specified with (19) [3]. Allowed large N , the near optimal estimate of the clock TIE can thus be obtained with the unbiased FIR filter (see an experimental verification given in [3]).
the averaging interval.
7.3. FIR Filtering Algorithms
7.2. Optimal FIR Filtering of TIE Utilizing N measurement points from n N + 1 to n , the FIR filtering estimate xˆn of xn is obtained, by (7) applied to (47), as
xˆn = WlT S n,m .
(63)
(57)
An important feature of FIR filtering is that it can be applied to each of the states separately [48]. Although, so far, we discussed the estimates of the clock first state xn , the remaining ones can be estimated similarly, even with individual N and , or all at once [3].
In order for the estimate (61) to be optimal, the degree of the gain Wl must be set such that l = K k , where K is the number of the states and k is the filtered state [14]. For the 3-state clock model and the first state filtered, one thus sets l=2.
7.3.1. Ladder Algorithm
The optimal gain matrix Wl 0 was found in [17, 21] for (61) to be
variance in this estimate is bounded with NPG g2 = a02 .
WlT0 = (A N 1 )1 R m CTnm (Z m + Z + V )1 ,
(58)
The ladder unbiased FIR filtering algorithm developed for the 3-state clock model is shown in Fig. (8) [14]. Here, the quadratic impulse response (26) is employed to produce the estimate xˆn of the TIE via the convolution (4). The noise
sn
where Z m = C nm R m CTnm and Z = G nm N GTnm . The noise covariance function matrices should be specialized in the averaging interval as
N = E{N n,m NTn,m },
(59)
V = E{Vn,m V }
(60)
T n,m
xˆn
h2 n xˆn xˆn 1
s yn
7.2.1. Large Averaging Intervals When the averaging interval occurs to be large, N >> 1, or the mean square initial state matrix Zm dominates in (62) the noise matrices in the order of magnitudes, then the second and third terms in the parentheses of (62) can be neglected and (62) simplifies to
WlT0 = (A N 1 )1 R m CTnm (C nm R m CTnm )1 .
yˆ n yˆ n 1
(61)
It has been shown in [3] that the initial state function Z m can a posteriori be determined in the filter algorithm by solving the discrete algebraic Riccati equation (DARE). Although (62) can be applied to the state space models with different kinds of noise sources (that is unlike the Kalman filter claiming all noise components be white sequences), it implies large computational burden when N is large. An essential simplification of (62) can be achieved when N >> 1 that is associated with GPS-based timekeeping and precision clocks.
(62)
Further multiplying the both sides of (66) with C nm R m CTnm and then removing nonsingular R m CTnm lead to the unbiasedness constraint
yˆ n
h1n
and the initial state covariance is
R m = E{x m xTm }.
szn
h0 n
zˆn
Fig. (8). The ladder FIR filtering algorithm for the 3-state clock model.
Because the second state, fractional frequency offset, can be specified by the discrete-time derivative of the TIE, the yn measurement is formed here as syn = ( xˆn xˆn1 ) / . Then the FIR filter with the ramp impulse response (25) is applied in order to estimate yˆn with the variance bounded with NPG
g1 = a01 . The procedure is repeated by applying simple averaging (24) to the derivative szn = ( yˆn yˆn1 ) / . That gives us an estimate zˆn of the linear frequency drift rate zn with the error variance subject to NPG g0 = 1 / N . For the 3 -state and 2 -state clock models, the ladder algorithms are listed below. 3-state clock model. If a clock is identified to have three states, the ladder algorithm is as follows:
xˆn =
N x 1
h
s
2i ni
i=0
,
(64)
144 Recent Patents on Space Technology, 2011, Volume 1, No. 2
yˆn =
zˆn =
1
Shmaliy and Ibarra-Manzano
N y 1
h
1j
N y 1
[ xˆn j xˆn j1 ],
(65)
y
j=0
1 Nz
yˆ k m =
N z 1
[ yˆ
nr
yˆnr1 ] ,
y
h s
1i yk y ( my 1)
i=0
.
sxn
(66)
r=0
(70)
where N x , N y , and N z are the individual averaging
s yk y m y
2-state clock model. For the 2-state clock model, the ladder algorithms implies N x 1
h
s
1i ni
,
yˆ k y k z mz yˆ k y k z ( mz 1)
(67)
1 Ny
N y 1
(xˆ
n j
xˆn j1 ).
s zk y k z m z
(68)
j=0
k z y z y
We have already mentioned, referring to [3], that xopt is 1 s for the first clock state in GPS-based measurements via the 1PPS signals. Unlike the first state, yopt and zopt must be set individually for the second and third states, respectively, in order to minimize errors. The relevant thinning algorithm has been proposed in [22] to operate in accordance with the diagram shown in Fig. (9). This algorithm accounts for the individual y and z , provided the thinning coefficients, ky and kz , such that information is not lost about the states. It follows from Fig. (9) that the first state xn is estimated by (68) as xˆn and that this estimate is then thinned out by
n changing to y = ky . In a new discrete scale my = , ky a where represents a maximum integer smaller than a/b, b the estimate becomes xˆ ky my . Next, the one pulse per ky seconds (1PP ky S) measurement of the second state is artificially formed by the backward time derivative as
xˆ ky my xˆ ky (my 1) k y
.
(69)
Further, the estimate yˆ k m of the second state is provided y
y
at each k y my point in a like manner using the gain h1i to
zˆk y k z mz
h0i ( N z )
n my k y
7.3.2. Thinning Algorithm
yield
yˆ k y k z mz
z
Here, the first accurate values of xˆn and yˆn appear at N x 1 and N x + N y 1 , respectively.
syky my =
yˆ k y m y
h1i ( N y )
i=0
yˆn =
k y y
y
at N x + N y + N z 1 .
xˆn =
xˆk y m y
xˆk y my xˆk y ( my 1)
intervals for the states. The first accurate value of xˆn appears yˆn zˆn at N x 1 , then at N x + N y 1 , and
xˆn
h2i ( N x )
n mz k k
y z
Fig. (9). The thinning FIR filtering algorithm for the 3-state clock model.
Similarly, the estimate zˆk k m y z
is found and one can
z
recognize two particular cases: 3-state clock model. For the clock identified to have three states, the thinning algorithm results in the following batch forms, xˆn =
N x 1
h
s
(71)
,
2i ni
i=0
yˆ k m = y
y
1 k y
zˆk k m = y z
z
N y 1
N z 1
h [ xˆ 1i
j=0
k y ( my j )
xˆk
y ( my
yˆ k k ( m r ) yˆ k k ( m r1) y z
z
y z
k y k z N z
r=0
z
.
j1)
],
(72)
(73)
It can be shown that the first correct estimate of the third state, for the worst case of asynchronous thinning, appears at N x 1 + k y ( N y + kz N z ) . 2-state clock model. The 2-state model fits well atomic clocks and is often applied to crystal clocks. For this model, the thinning algorithm is as follows xˆn =
N x 1
h s
1i ni
i=0
,
(74)
GPS-Based FIR Estimation and Steering of Clock Errors
1 k y N y
yˆ k m = y
y
Recent Patents on Space Technology, 2011, Volume 1, No. 2
N y 1
[ xˆk j=0
y
xˆk (m j) y
y
]. ( m j1)
(75)
y
In the worst case of asynchronous thinning, the first correct estimate of the second state appears at Nx 1 + ky N y . Applications of both ladder and thinning algorithms to crystal clocks can be found in [13-24]. 7.3.3. Generic p-Shift Optimal Estimation Algorithm If to assign the gain matrix H( p)
K N
(81)
that certainly implies the computational problem when N is large, N >> 1 . To compute the estimate iteratively, the relevant Kalman-like algorithm was derived in [3] as shown in Table 2. Here, a set of auxiliary functions, , PK 1 , and
FK 1 , is first computed and the initial vector x n N + K + p
determined a posteriori via measurement without prediction. Then an auxiliary matrix Fv and the estimate vector x n N +1+v+ p are both updated by changing v from K to
, then the p-
shift estimate xn+ p of the clock state x n can be found at
n+ p
= A nm+ p (CTnmC nm )1 CTnmS n,m
145
as x n+ p = H( p)S n,m . It has been shown in [3] that the
N 1 for all n K. The true estimate is taken when v = N 1. Table 2.
Iterative Kalman-like Estimation Algorithm
p-Shift
Unbiased
FIR
relevant optimal estimate has two equivalent forms of + )1 S x n+ p = H( p)Z m (Z m + Z V n,m
(76)
Stage
+ )1 S , = A nm+ p (CTnm C nm )1 CTnm Z m (Z m + Z V n,m
(77)
Given:
where Z m is determined by solving the discrete algebraic
Set:
n,m
n,m
V
FK 1 = A K 1PK 1 (A K 1 )T x n N + K + p = A K 1+ p P CT S
(78)
m
K 1
Update:
and H( p) = A nm+ p (CTnmC nm )1 CTnm
V = 0 . Table 1 summarizes the p -shift optimal FIR
estimation algorithm of the clock state at n + p . Table 1.
Generic p-Shift Optimal FIR Estimation Algorithm
Stage Given:
N K and p
Set:
H( p) = A N 1+ p (CTN 1C N 1 )1 CTN 1
Find Z n N +1 :
0 = Z n N +1 (Z + V )1 Z n N +1 + 2Z n N +1 + Z + V Sn,n N +1STn,n N +1 (Z + V )1 Z n N +1
Compute
n ( p) = H( p)Z n N +1 (Z n N +1 + Z + V )1 H ( p)S x n+ p = H n n,n N +1
7.3.4. Iterative Kalman-Like Estimation Algorithm
p-Shift
Unbiased
FIR
It follows from (83) that the unbiased FIR estimate of clock state can be found in the batch form as
x n + p|n = H ( p )S n ,m
(80)
K 1n N + K ,n N +1
Fv = AFv1 AT AFv1 (I + Fv1 )1 Fv1 AT x n N +1+v+ p = Ax n N +v+ p + A p F CT (sn N +1+v CA1 p x n N +v+ p )
(79)
represents the unbiased gain subject to the unbiasedness = 0 and condition (12) or to the deterministic case of Z
= AT CT CA PK 1 = (CTK 1C K 1 )1
Riccati equation (DARE)
0 = Z m (Z + V )1 Z m + 2Z m + Z + V S ST (Z + )1 Z
K, N , p, v = K,..., N 1
Instruction:
v
Use x n N +1+v+ p as the output when v = N 1
An important advantage of this algorithm is that it has the Kalman form, but does not require neither the noise covariance nor the initial error. However, iterations consume about N times more computation time that can be circumvented with parallel computing. Note that both algorithms (Tables 1 and 2) can be applied with the fixed or variable N. In a special case when all the data available need to be processed, the relevant full-horizon algorithms imply setting the initial point to zero and substituting N with n + 1, as shown in [28]. 8. THE TRADE-OFF ALGORITHM
WITH
THE
KALMAN
The Kalman algorithm is often employed in clock synchronization and timescales in the white Gaussian approximation as shown in [3]. Its advantage resides in fast and accurate estimates if the model is not disturbed substantially by the clock colored noise, non-Gaussian measurement noise, and GPS time temporary uncertainties. Otherwise, the recursive Kalman computation may produce redundant errors. Contrary to the infinite impulse response (IIR) recursive structures, such as the one of the Kalman filter, the transversal FIR structures have several important properties critical for clocks:
146 Recent Patents on Space Technology, 2011, Volume 1, No. 2
Shmaliy and Ibarra-Manzano
•
Noise having any distribution and known covariance function is allowed. So, noise should not obligatorily be white sequence as Kalman claims.
•
Inherent BIBO stability featured to averaging filters.
•
Better robustness uncertainties.
against
temporary
behavior). Note that the filters outputs represent well the GPS time structure (temporary uncertainty) caused by different satellites in a view. –150
TIE, ns
model
GPS measurement –200
•
Better robustness against round-off errors, especially when N >> 1.
•
Convergence of optimal and unbiased estimates with N >> 1.
•
Low sensitivity to noise featured to averaging.
Extensive investigations of the Kalman and FIR filters in applications to different kinds of clocks can be found in [1324]. As an immediate example, Fig. (10) sketches typical errors in the estimates of the first state provided with the 2degree (ramp) optimal FIR filter and the two-state Kalman filter, both applied to the GPS-based TIE measurement of the crystal clock imbedded in the Stanford Frequency Counter SR620. As can be seen, there is no time delay between the estimates and the filters thus have similar time constants. Herewith, the FIR filter demonstrates better robustness against the GPS time temporary uncertainty, producing smaller noise and lower regular errors.
Optimal FIR (white)
–250 –300
–350
Kalman (black)
Actual TIE 0.5
1
1.5 Time, hours
2
2.5
(a) –5
10
TDEV, ns
–6
10
GPS measurement
–7
Optimal FIR
10
–8
10
15
Kalman
Kalman –9
10
10
–10
0
10
Optimal FIR
5
2000
3000
1000
–5
Actual
–11
4000
10
–12
Time, n
Fig. (10). Typical errors in the GPS-based TIE filtering of the 1state of a crystal clock, after [21]. Estimates are obtained by the 2degree optimal FIR filter and 2-state Kalman filter.
9. APPLICATIONS Below, we observe several typical applications of the FIR and Kalman filtering to GPS-based timekeeping and clock estimation. The results have been obtained in the set (Fig. 1) with the GPS timing SynPaQ III Sensor as a receiver, Stanford Frequency Counter SR625, and local crystal clock imbedded in another SR625. The actual clock errors have been measured for the Symmetricom cesium standard of frequency CsIII. 9.1. TIE Estimation Fig. (11a) sketches a typical picture of the GPS-based measurement with sawtooth (for about 2 hours) of the clock TIE along with the actual TIE behavior. It also incorporates the unbiased FIR estimate xˆn and the Kalman one xˆ Kn . As can instantly be deduced, both estimates are consistent, unbiased, and noisy (as compared to the actual TIE
10
10
0
10
1
2
10 , s (b)
10
3
10
4
Fig. (11). Typical GPS-based measurement with sawtooth and estimates of the TIE of a crystal clock, after [22]: (a) measurements and estimates, and (b) TDEV.
To ascertain the difference between the estimates, Fig. (11b) shows the time deviation (TDEV) [60, 61] computed over all measured points. As can be seen, the TDEV associated with xˆn and xˆ Kn behave closely to each other and trace above the actual values, although, a bit better behavior of the FIR estimate can be pointed out. 9.2. Estimation of the Frequency Offset In [22], the thinning algorithm (79) was applied in order to estimate the second clock state. Following [16], the minimum MSE in the FIR estimate was found in the range of 10 s < < 100 s and the filter parameters were allowed to be k y = 100 and N y = 130 . Fig. (12) shows the relevant GPSbased and reference measurements along with the FIR and Kalman estimates.
GPS-Based FIR Estimation and Steering of Clock Errors
3.0
Recent Patents on Space Technology, 2011, Volume 1, No. 2
147
been measured to range from -30ns to 40ns over time as shown in Fig. (13a). An additional 1-order low-pass (LP) filter included to the loop with time constant T = 2000 s was used to smooth the TIE as shown in Fig. (13b).
Fractional frequency offset, 10-10
2.5 GPS measurement 2.0 Kalman
1.5
40 30 20 10 0 –10 –20 –30
Actual 1.0 0.5
Optimal FIR ky = 100 Ny = 130
TIE, ns
Sawtooth, N = 250 80
20 40
60 Time, hours (a)
0 –0.5
10
15
20
25
35 30 Time, hours (a)
45
40
10–7
50
Crystal Clock in SR620
10–8
GPS measurement ADEV Optimal FIR ky = 100 Ny = 130
10–9 Kalman 10–10
10–11 Actual 10–12 100
101
102 , s (b)
103
104
Fig. (12). Typical GPS-based measurement with sawtooth and estimates of the fractional frequency offset of a crystal clock, after [22]: (a) measurements and estimates, and (b) ADEV.
As can be observed (Fig. 12a), the FIR estimate xˆ2 and the Kalman one xˆ2 K fit well both the actual trend (reference measurement) and the time derivative of the thinned out estimate of the TIE (GPS-measurement). The goodness-of-fit estimates are demonstrated in Fig. (12b) in terms of the Allan deviation (ADEV) [16]. One notices that the FIR filter produces much lower noise that the Kalman filter when
< 103 s, although both estimates still trace above the actual value. 9.3. Clock Steering Steering the local clock errors in the loop (Fig. 1) has been provided in [13]. Because only fast noise components are commonly filtered out in clock synchronization, the averaging interval was chosen as shown in Fig. (2, “Filtering of TIE”) to produce the minimum MSE in the output of a GPS-locked crystal clock. For the GPS-based measurement with sawtooth, the optimum N was experimentally found, following [16], to be N opt 250 . By this value, TIE has
40 30 20 10 0 –10 –20 –30
TIE, ns
Sawtooth, N = 250, 1-order LP Filter, = 2000 s 20 40
60 Time, hours (b)
80
Fig. (13). Typical errors in the GPS-based TIE filtering estimates produced by the 2-degree optimal FIR filter and 2-state Kalman filter, after [19].
As can be seen, the TIE smoothed (Fig. 13b) ranges within the same bounds as the origin (Fig. 13a). Therefore, a more subtle comparison of both functions had been provided in [13] in terms of the TDEV and ADEV (Fig. 14). This result suggests that TDEV of the unlocked crystal clock inherently increases with large averaging time, whereas the reference GPS 1PPS signal with sawtooth has a uniform TDEV function (Fig. 14a). The optimal FIR filter included to the loop allows for an efficient steering of clock errors with large averaging times. An additional smoothing filter (1-order LP, T = 1000 s) completes the picture, providing a substantial noise reduction at small averaging times. It can be seen that the resulting TDEV traces much lower the mask specified in [61] for digital communications networks. Contrary to TDEV, the ADEV of the reference GPS 1PPS signal decreases linearly in the Bode plot, whereas that of the unlocked crystal clock inherently reduces and then grows passing through a minimum (Fig. 14b). The effect of the optimal FIR filter with the smoothing 1-order LP filter included in the loop results in the following. In the locked crystal clock, noise with small averaging times mostly depends on precision of its oscillator (effect of the smoothing filter). The time error departures with large averaging times are limited here by inaccuracy of the optimal FIR filter. Note that an excursion in the ADEV observed in the middle of the averaging times range shown in Fig. (14) cannot be avoided from the standpoint of control. Its peak value can only be minimized and position optimized by smoothing. 10. CONCLUSIONS In this paper, we referred to recent patents and observed advances in clock estimation and synchronization with applications to GPS-based timekeeping. Advantages of the optimal FIR approach against the Kalman filter were shown experimentally for real measurements in the presence
148 Recent Patents on Space Technology, 2011, Volume 1, No. 2
Shmaliy and Ibarra-Manzano [5]
10–5
N = 250
TDEV, ns
[6]
10–6
Unlocked Crystal Clock
[7]
10–7
GPS 1PPS
10–8
[8]
FIR Filter
ITU-T G.811
[9]
10–9 Locked Crystal Clock
FIR Filter + LP Filter T = 1000 s
10–10
[10] 10
–11
100
101
102 103 Averaging time, s (a)
104
105
[11]
[12] 10–7
N = 250
10–8 10
[13] GPS 1PPS
–9
FIR Filter Unlocked Crystal Clock
10–10
[14]
Locked Crystal Clock
10–11
[15]
10–12
FIR Filter + LP Filter T = 2000 s
ADEV 10–13 0 10
101
102
[16] 103
Averaging time, s
104
105
(b)
Fig. (14). Typical steering errors of the GPS-locked crystal clock, after [13]: (a) TDEV, and (b) ADEV.
of the GPS time temporary uncertainty and non-Gaussian sawtooth noise induced by the GPS timing receiver. It was demonstrated that the unbiased FIR algorithms (filters, predictors, and smoothers) serve better in GPS-based timekeeping than the Kalman one in terms of accuracy, stability, and robustness. A payment for these advantages is about N times larger consumption of the computation time featured to the averaging. This payment can easily be avoided in parallel computing of FIR estimates. We finally emphasize once again the extremely strong features of the optimal FIR filter against the Kalman one: 1) its algorithm is able to produce optimal estimates in arbitrary noise environment with no requirements for the initial error and 2) its unbiased version becomes virtually optimal when N >> 1. Note that both these situations are typical for clocks. REFERENCES [1] [2]
[3] [4]
D.W. Allan, N. Ashby, and C.C. Hodge, “The Science of Timekeeping,” Application Note 1289. Hewlett Packard: Englewood, CO, pp 1-88, 1997. D.W. Allan, and J.A. Barnes, “Optimal time and frequency transfer using GPS signals,” in 36th Annual Frequency Control Symposium, Philadelphia, PA, 1982, pp. 378-387. Y.S. Shmaliy, GPS-Based Optimal FIR Filtering of Clock Models, New York: Nova Science Publ., 2009. R.E. Kalman, “A new approach to linear filtering and prediction problems,” J. Basic Engin., vol. 82, pp. 35-45, March 1960.
[17] [18]
[19] [20]
[21]
[22]
[23] [24]
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